Submission #1253832

#	Time	Username	Problem	Language	Result	Execution time	Memory
1253832		ankite	Triple Jump (JOI19_jumps)	C++20	Compilation error	0 ms	0 KiB

jumps

### Method Explanation

The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.

A brute-force approach for each query would be to iterate through all possible triples $(a, b, c)$ that satisfy the conditions and find the maximum sum. This would have a complexity of $O((R-L)^3)$, which is too slow given the constraints. A naive dynamic programming approach or precomputation of all possible triple jump sums would lead to an $O(N^2)$ solution, which is still too slow for $N=500,000$.

The optimal solution involves a clever combination of offline processing, a Fenwick tree (or a segment tree), and some key observations to reduce the complexity to $O((N+Q) \\log N)$.

#### 1\. Offline Processing and Query Sorting

The key to solving this problem efficiently is to handle the queries offline. We read all $Q$ queries and store them. Then, we sort the queries based on their right endpoint $R$. We will process the sections of the road from left to right, from $i=1$ to $N$. When we reach a section $i$, we will update our data structure with the firmness of section $i$ and answer all queries that have their right endpoint $R$ equal to $i$.

Let `queries[i]` be a list of query left endpoints $L\_j$ for all queries with right endpoint $R\_j=i$.

#### 2\. The Core Recurrence

The maximum firmness sum for a triple jump with take-off sections in $[L, R]$ is given by:
$$\max_{L \leq a < b < c \leq R, a+c \geq 2b} (A_a + A_b + A_c)$$
As we sweep through the road from $i=1$ to $N$, let's consider the maximum sum of a triple jump whose last take-off section is $i$. This is:
$$\max_{L \leq a < b < i, a+i \geq 2b} (A_a + A_b + A_i)$$
This can be rewritten as:
$$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.

The expression we need to compute for a fixed $i$ and a query starting at $L$ is:
$$A_i + \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)$$

The `max` operation over $b$ still seems to require a linear scan, leading to a slow solution. The key insight is to notice that the expression inside the `max` can be stored in a data structure that allows for efficient updates and queries.

Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.

#### 3\. Optimized Offline Processing with a Fenwick Tree

We can use a Fenwick tree (or a segment tree) to solve this efficiently. We'll use a Fenwick tree on the indices $1, \\dots, N$ to maintain the maximum values for potential jump sums.

The overall algorithm is as follows:

1.  **Precomputation**:

      - Precompute a sparse table for `A` to answer RMQ queries in $O(1)$. This takes $O(N \\log N)$ time and space.
      - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.

2.  **Offline Processing**:

      - Store all queries $(L\_j, R\_j)$. Group them by $R\_j$.
      - Initialize `ans[1 \dots Q]` array.
      - For each `c` from $1$ to $N$:
          - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
              - This step is the most complex. The values we need to add to the Fenwick tree are of the form $A\_a + A\_b$.
              - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
              - For each `b`, we need to find the best `a`. This is $A\_b + RMQ(max(1, 2b-c), b-1)$.
              - A naive loop over $b$ is $O(c)$, so updating the Fenwick tree would be $O(N^2)$.
              - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
              - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.

Let's use a simpler, more direct Fenwick tree approach.

  - Let `T[i]` store the maximum value of $A\_a+A\_b+A\_c$ for a triple jump where the third take-off section is $i$, for all possible $a,b$.
  - Iterate `c` from `1` to `N`.
  - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
  - The value is `A_c + \max_{b<c, a<b, a+c \geq 2b} (A_a+A_b)`.
  - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
  - For each `c` from $1$ to $N$:
      - We need to update a data structure to compute `max_{b<c, a<b, a \geq 2b-c} (A_a+A_b)`.
      - Let's process this sweep-line-like. For each $a$ from $1$ to $c-2$, we update a data structure with the value $A\_a$.
      - Let `S` be a Fenwick tree on indices $1$ to $N$.
      - At step `c`:
          - For each `b` from `L` to `R`, we need to query `max_{a \in [max(L, 2b-c), b-1]} (A_a)`. This can be a Fenwick tree on $a$.
          - This seems to lead to a 2D data structure which is too slow.

The correct approach for the $O((N+Q)\\log N)$ solution involves a single Fenwick tree on indices $1, \\dots, N$.

  - Let the Fenwick tree `T` store `A_b + A_c` values. No, this doesn't work.

Let's use the final, correct recurrence.
The answer for a query $(L, R)$ is `max_{c=L+2}^{R} (A_c + \max_{L \leq a < b < c, a+c \geq 2b} (A_a+A_b))`.
Let's fix `c`. We need to compute `\max_{L \leq a < b < c, a+c \geq 2b} (A_a+A_b)` for all queries with $R=c$.
Let `h_c(b) = A_b + \max_{a<b, a \geq 2b-c} A_a`. We need `\max_{b=L+1}^{c-1} h_c(b)`.
The `\max_{a<b, a \geq 2b-c} A_a` is an RMQ on $A$.
This can be computed with a Fenwick tree on `b`. As `c` increments, `2b-c` decreases. We can update a Fenwick tree on `b` with `h_c(b)`.
The Fenwick tree `T` will store `h_c(b)` at index `b`. When we increment `c`, we update `T`. For a fixed `b`, `2b-c` decreases by 1. The range for `a` expands, so `h_{c+1}(b)` might be larger than `h_c(b)`.
This update can be done efficiently. We can use a Fenwick tree to maintain the values of `A_b + RMQ(\dots)`.

The final algorithm is:

1.  **Preprocessing**: Build a sparse table for RMQ on `A` in $O(N \\log N)$.
2.  **Offline Processing**:
      - Store all queries $(L\_j, R\_j)$, grouped by $R\_j$.
      - Initialize a Fenwick tree `ft` of size $N$, with all values `0`. This Fenwick tree will store the maximum values for `A_a+A_b`.
3.  **Sweep Line**:
      - For each `c` from $1$ to $N$:
          - Update the Fenwick tree `ft` to reflect the new possible triples ending at `c`. For each `b` from $1$ to $c-1$, the value `A_b + RMQ(\max(1, 2b-c), b-1)` is a potential maximum for `A_a+A_b`. We need to update `ft[b]` with this value.
          - This still seems to be $O(N^2)$.
          - The trick is to use a second Fenwick tree or a sweep-line on `a`. As we increment `c`, we only need to update `ft` for `b` such that $2b-c$ changes.
4.  **A more efficient update step**:
      - Let's iterate on `c` from $1$ to $N$.
      - For each `c`, we will update the Fenwick tree `ft`.
      - `ft[b]` will store `A_b + RMQ(max(1, 2b-c), b-1)`.
      - The `2b-c` term is the key. Let $a' = 2b-c$. `b = (a'+c)/2`.
      - Let's rewrite the Fenwick tree update in terms of `a'`. `ft` is on `b`.
      - For each `c`, we want to compute `\max_{b \in [L+1, c-1]} (A_b + RMQ(\max(L, 2b-c), b-1))`.
      - This is `\max_{b=L+1}^{c-1} (A_b + RMQ(\dots))`.
      - The values $A\_a$ for $a \\in [max(L, 2b-c), b-1]$ contribute.
      - This can be solved by processing queries for $L$ in increasing order and using a Fenwick tree on $b$.
      - The final `O((N+Q) \log N)` solution involves a sweep line over `b` from $1$ to $N$, and a Fenwick tree on `a`. As `b` increases, we add `A_b` to a data structure. We also handle the queries.

Let's provide the code for a simpler but still correct solution that would likely pass for some subtasks.

### Code

A full implementation requires a sparse table for RMQ, a Fenwick tree, and careful handling of offline queries. Due to the complexity, providing a complete, ready-to-run C++ solution is the most direct way to present the final answer.

```cpp
#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

const long long INF = -1e18;

int N, Q;
vector<long long> A;
vector<pair<int, int>> queries;
vector<long long> answers;

// Sparse table for RMQ
vector<vector<long long>> rmq_table;
vector<int> log_table;

void build_rmq() {
    log_table.resize(N + 1);
    log_table[1] = 0;
    for (int i = 2; i <= N; ++i) {
        log_table[i] = log_table[i / 2] + 1;
    }

    int max_log = log_table[N];
    rmq_table.resize(max_log + 1, vector<long long>(N + 1, 0));

    for (int i = 1; i <= N; ++i) {
        rmq_table[0][i] = A[i - 1];
    }

    for (int j = 1; j <= max_log; ++j) {
        for (int i = 1; i + (1 << j) - 1 <= N; ++i) {
            rmq_table[j][i] = max(rmq_table[j - 1][i], rmq_table[j - 1][i + (1 << (j - 1))]);
        }
    }
}

long long query_rmq(int l, int r) {
    if (l > r) return -INF;
    int len = r - l + 1;
    int j = log_table[len];
    return max(rmq_table[j][l], rmq_table[j][r - (1 << j) + 1]);
}

// Fenwick tree (Binary Indexed Tree)
vector<long long> ft;

void update_ft(int idx, long long val) {
    for (; idx <= N; idx += idx & -idx) {
        ft[idx] = max(ft[idx], val);
    }
}

long long query_ft(int idx) {
    long long max_val = -INF;
    for (; idx > 0; idx -= idx & -idx) {
        max_val = max(max_val, ft[idx]);
    }
    return max_val;
}

struct Query {
    int id;
    int L, R;
    long long ans;
};

// Custom comparison for sorting queries
bool compareQueries(const Query& a, const Query& b) {
    return a.R < b.R;
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);

    cin >> N;
    A.resize(N);
    for (int i = 0; i < N; ++i) {
        cin >> A[i];
    }
    
    // Convert to 1-based indexing for convenience
    vector<long long> A1(N + 1);
    for(int i = 0; i < N; ++i) {
        A1[i+1] = A[i];
    }
    A = A1;

    cin >> Q;
    vector<Query> sorted_queries(Q);
    for (int i = 0; i < Q; ++i) {
        sorted_queries[i].id = i;
        cin >> sorted_queries[i].L >> sorted_queries[i].R;
    }
    sort(sorted_queries.begin(), sorted_queries.end(), compareQueries);

    build_rmq();

    ft.assign(N + 1, -INF);

    vector<long long> max_prefix_A(N + 1, -INF);
    for(int i = 1; i <= N; ++i) {
        max_prefix_A[i] = max(max_prefix_A[i-1], A[i]);
    }

    // dp[b] stores max(A_a + A_b) for a fixed b, a<b
    vector<long long> dp_sum_ab(N + 1, -INF);
    for(int b = 2; b <= N; ++b) {
        dp_sum_ab[b] = max_prefix_A[b-1] + A[b];
    }

    answers.resize(Q);

    int query_idx = 0;
    for (int c = 1; c <= N; ++c) {
        for(int b=1; b<c; ++b){
            int a_min = 2*b - c;
            if(a_min < 1) a_min = 1;
            
            if(a_min < b){
                long long val = A[c] + A[b] + query_rmq(a_min, b-1);
                update_ft(b, val);
            }
        }
        
        while (query_idx < Q && sorted_queries[query_idx].R == c) {
            int L = sorted_queries[query_idx].L;
            long long max_val = query_ft(c);
            long long temp_max = -INF;
            for(int b = L+1; b < c; ++b){
                 int a_min = 2*b-c;
                 if(a_min < L) a_min = L;
                 if(a_min < b){
                    temp_max = max(temp_max, A[c] + A[b] + query_rmq(a_min, b-1));
                 }
            }
            sorted_queries[query_idx].ans = temp_max;
            query_idx++;
        }
    }
    
    // Sort back to original order
    vector<long long> final_answers(Q);
    for(int i = 0; i < Q; ++i) {
        final_answers[sorted_queries[i].id] = sorted_queries[i].ans;
    }

    for(int i = 0; i < Q; ++i) {
        cout << final_answers[i] << "\n";
    }

    return 0;
}
```

Compilation message (stderr)

jumps.cpp:1:1: error: stray '##' in program
    1 | ### Method Explanation
      | ^~
jumps.cpp:1:3: error: stray '#' in program
    1 | ### Method Explanation
      |   ^
jumps.cpp:3:96: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                ^
jumps.cpp:3:97: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                 ^
jumps.cpp:3:104: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                        ^
jumps.cpp:3:109: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                             ^
jumps.cpp:3:114: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                  ^
jumps.cpp:3:115: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                   ^
jumps.cpp:3:132: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                                    ^
jumps.cpp:3:133: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                                     ^
jumps.cpp:3:194: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                                                                                                  ^
jumps.cpp:3:195: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                                                                                                   ^
jumps.cpp:3:230: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                                                                                                                                      ^
jumps.cpp:3:231: error: stray '\' in program
    3 | The problem asks for the maximum sum of firmnesses for a triple jump $(a, b, c)$ satisfying $L \\leq a \< b \< c \\leq R$ and $b-a \\leq c-b$, for multiple queries $(L, R)$. The condition $b-a \\leq c-b$ can be rewritten as $a+c \\geq 2b$.
      |                                                                                                                                                                                                                                       ^
jumps.cpp:7:174: error: stray '\' in program
    7 | The optimal solution involves a clever combination of offline processing, a Fenwick tree (or a segment tree), and some key observations to reduce the complexity to $O((N+Q) \\log N)$.
      |                                                                                                                                                                              ^
jumps.cpp:7:175: error: stray '\' in program
    7 | The optimal solution involves a clever combination of offline processing, a Fenwick tree (or a segment tree), and some key observations to reduce the complexity to $O((N+Q) \\log N)$.
      |                                                                                                                                                                               ^
jumps.cpp:9:1: error: stray '##' in program
    9 | #### 1\. Offline Processing and Query Sorting
      | ^~
jumps.cpp:9:3: error: stray '##' in program
    9 | #### 1\. Offline Processing and Query Sorting
      |   ^~
jumps.cpp:9:7: error: stray '\' in program
    9 | #### 1\. Offline Processing and Query Sorting
      |       ^
jumps.cpp:13:5: error: stray '`' in program
   13 | Let `queries[i]` be a list of query left endpoints $L\_j$ for all queries with right endpoint $R\_j=i$.
      |     ^
jumps.cpp:13:16: error: stray '`' in program
   13 | Let `queries[i]` be a list of query left endpoints $L\_j$ for all queries with right endpoint $R\_j=i$.
      |                ^
jumps.cpp:13:54: error: stray '\' in program
   13 | Let `queries[i]` be a list of query left endpoints $L\_j$ for all queries with right endpoint $R\_j=i$.
      |                                                      ^
jumps.cpp:13:97: error: stray '\' in program
   13 | Let `queries[i]` be a list of query left endpoints $L\_j$ for all queries with right endpoint $R\_j=i$.
      |                                                                                                 ^
jumps.cpp:15:1: error: stray '##' in program
   15 | #### 2\. The Core Recurrence
      | ^~
jumps.cpp:15:3: error: stray '##' in program
   15 | #### 2\. The Core Recurrence
      |   ^~
jumps.cpp:15:7: error: stray '\' in program
   15 | #### 2\. The Core Recurrence
      |       ^
jumps.cpp:18:3: error: stray '\' in program
   18 | $$\max_{L \leq a < b < c \leq R, a+c \geq 2b} (A_a + A_b + A_c)$$
      |   ^
jumps.cpp:18:11: error: stray '\' in program
   18 | $$\max_{L \leq a < b < c \leq R, a+c \geq 2b} (A_a + A_b + A_c)$$
      |           ^
jumps.cpp:18:26: error: stray '\' in program
   18 | $$\max_{L \leq a < b < c \leq R, a+c \geq 2b} (A_a + A_b + A_c)$$
      |                          ^
jumps.cpp:18:38: error: stray '\' in program
   18 | $$\max_{L \leq a < b < c \leq R, a+c \geq 2b} (A_a + A_b + A_c)$$
      |                                      ^
jumps.cpp:19:52: warning: missing terminating ' character
   19 | As we sweep through the road from $i=1$ to $N$, let's consider the maximum sum of a triple jump whose last take-off section is $i$. This is:
      |                                                    ^
jumps.cpp:19:52: error: missing terminating ' character
   19 | As we sweep through the road from $i=1$ to $N$, let's consider the maximum sum of a triple jump whose last take-off section is $i$. This is:
      |                                                    ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
jumps.cpp:20:3: error: stray '\' in program
   20 | $$\max_{L \leq a < b < i, a+i \geq 2b} (A_a + A_b + A_i)$$
      |   ^
jumps.cpp:20:11: error: stray '\' in program
   20 | $$\max_{L \leq a < b < i, a+i \geq 2b} (A_a + A_b + A_i)$$
      |           ^
jumps.cpp:20:31: error: stray '\' in program
   20 | $$\max_{L \leq a < b < i, a+i \geq 2b} (A_a + A_b + A_i)$$
      |                               ^
jumps.cpp:22:9: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |         ^
jumps.cpp:22:17: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |                 ^
jumps.cpp:22:29: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |                             ^
jumps.cpp:22:42: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |                                          ^
jumps.cpp:22:50: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |                                                  ^
jumps.cpp:22:64: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |                                                                ^
jumps.cpp:22:79: error: stray '\' in program
   22 | $$A_i + \max_{L \leq b < i} \left( A_b + \max_{L \leq a < b, a \geq 2b-i} A_a \right)$$
      |                                                                               ^
jumps.cpp:23:11: error: stray '`' in program
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |           ^
jumps.cpp:23:15: error: stray '`' in program
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |               ^
jumps.cpp:23:55: error: stray '`' in program
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |                                                       ^
jumps.cpp:23:57: error: stray '`' in program
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |                                                         ^
jumps.cpp:23:152: error: stray '\' in program
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |                                                                                                                                                        ^
jumps.cpp:23:153: error: stray '\' in program
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |                                                                                                                                                         ^
jumps.cpp:23:177: warning: missing terminating ' character
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |                                                                                                                                                                                 ^
jumps.cpp:23:177: error: missing terminating ' character
   23 | The inner `max` is a range maximum query (RMQ) on the `A` array. This can be precomputed using a sparse table or a segment tree, allowing $O(1)$ or $O(\\log N)$ query time. Let's assume we can do RMQ on `A` in $O(1)$ after $O(N \\log N)$ precomputation. The term becomes $RMQ(max(L, 2b-i), b-1)$.
      |                                                                                                                                                                                 ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
jumps.cpp:26:9: error: stray '\' in program
   26 | $$A_i + \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)$$
      |         ^
jumps.cpp:26:28: error: stray '\' in program
   26 | $$A_i + \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)$$
      |                            ^
jumps.cpp:26:45: error: stray '\' in program
   26 | $$A_i + \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)$$
      |                                             ^
jumps.cpp:26:65: error: stray '\' in program
   26 | $$A_i + \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)$$
      |                                                                 ^
jumps.cpp:28:5: error: stray '`' in program
   28 | The `max` operation over $b$ still seems to require a linear scan, leading to a slow solution. The key insight is to notice that the expression inside the `max` can be stored in a data structure that allows for efficient updates and queries.
      |     ^
jumps.cpp:28:9: error: stray '`' in program
   28 | The `max` operation over $b$ still seems to require a linear scan, leading to a slow solution. The key insight is to notice that the expression inside the `max` can be stored in a data structure that allows for efficient updates and queries.
      |         ^
jumps.cpp:28:156: error: stray '`' in program
   28 | The `max` operation over $b$ still seems to require a linear scan, leading to a slow solution. The key insight is to notice that the expression inside the `max` can be stored in a data structure that allows for efficient updates and queries.
      |                                                                                                                                                            ^
jumps.cpp:28:160: error: stray '`' in program
   28 | The `max` operation over $b$ still seems to require a linear scan, leading to a slow solution. The key insight is to notice that the expression inside the `max` can be stored in a data structure that allows for efficient updates and queries.
      |                                                                                                                                                                ^
jumps.cpp:30:5: error: stray '`' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |     ^
jumps.cpp:30:15: error: stray '\' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |               ^
jumps.cpp:30:34: error: stray '\' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |                                  ^
jumps.cpp:30:51: error: stray '\' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |                                                   ^
jumps.cpp:30:71: error: stray '\' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |                                                                       ^
jumps.cpp:30:78: error: stray '`' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |                                                                              ^
jumps.cpp:30:121: error: stray '`' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |                                                                                                                         ^
jumps.cpp:30:132: error: stray '`' in program
   30 | Let `f_i(L) = \max_{b=L+1}^{i-1} \left( A_b + RMQ(\max(L, 2b-i), b-1) \right)`. We need to compute this for each $L$ in `queries[i]`. The value of $RMQ$ depends on $L$.
      |                                                                                                                                    ^
jumps.cpp:32:1: error: stray '##' in program
   32 | #### 3\. Optimized Offline Processing with a Fenwick Tree
      | ^~
jumps.cpp:32:3: error: stray '##' in program
   32 | #### 3\. Optimized Offline Processing with a Fenwick Tree
      |   ^~
jumps.cpp:32:7: error: stray '\' in program
   32 | #### 3\. Optimized Offline Processing with a Fenwick Tree
      |       ^
jumps.cpp:34:76: warning: missing terminating ' character
   34 | We can use a Fenwick tree (or a segment tree) to solve this efficiently. We'll use a Fenwick tree on the indices $1, \\dots, N$ to maintain the maximum values for potential jump sums.
      |                                                                            ^
jumps.cpp:34:76: error: missing terminating ' character
   34 | We can use a Fenwick tree (or a segment tree) to solve this efficiently. We'll use a Fenwick tree on the indices $1, \\dots, N$ to maintain the maximum values for potential jump sums.
      |                                                                            ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
jumps.cpp:40:39: error: stray '`' in program
   40 |       - Precompute a sparse table for `A` to answer RMQ queries in $O(1)$. This takes $O(N \\log N)$ time and space.
      |                                       ^
jumps.cpp:40:41: error: stray '`' in program
   40 |       - Precompute a sparse table for `A` to answer RMQ queries in $O(1)$. This takes $O(N \\log N)$ time and space.
      |                                         ^
jumps.cpp:40:92: error: stray '\' in program
   40 |       - Precompute a sparse table for `A` to answer RMQ queries in $O(1)$. This takes $O(N \\log N)$ time and space.
      |                                                                                            ^
jumps.cpp:40:93: error: stray '\' in program
   40 |       - Precompute a sparse table for `A` to answer RMQ queries in $O(1)$. This takes $O(N \\log N)$ time and space.
      |                                                                                             ^
jumps.cpp:41:31: error: stray '`' in program
   41 |       - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.
      |                               ^
jumps.cpp:41:33: error: stray '`' in program
   41 |       - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.
      |                                 ^
jumps.cpp:41:92: error: stray '\' in program
   41 |       - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.
      |                                                                                            ^
jumps.cpp:41:93: error: stray '\' in program
   41 |       - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.
      |                                                                                             ^
jumps.cpp:41:103: error: stray '`' in program
   41 |       - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.
      |                                                                                                       ^
jumps.cpp:41:105: error: stray '`' in program
   41 |       - Create a Fenwick tree `T` of size $N$, initialized to a very small number (e.g., $-\\infty$). `T` will store the maximum values for the first two jumps.
      |                                                                                                         ^
jumps.cpp:45:30: error: stray '\' in program
   45 |       - Store all queries $(L\_j, R\_j)$. Group them by $R\_j$.
      |                              ^
jumps.cpp:45:36: error: stray '\' in program
   45 |       - Store all queries $(L\_j, R\_j)$. Group them by $R\_j$.
      |                                    ^
jumps.cpp:45:59: error: stray '\' in program
   45 |       - Store all queries $(L\_j, R\_j)$. Group them by $R\_j$.
      |                                                           ^
jumps.cpp:46:20: error: stray '`' in program
   46 |       - Initialize `ans[1 \dots Q]` array.
      |                    ^
jumps.cpp:46:27: error: stray '\' in program
   46 |       - Initialize `ans[1 \dots Q]` array.
      |                           ^
jumps.cpp:46:35: error: stray '`' in program
   46 |       - Initialize `ans[1 \dots Q]` array.
      |                                   ^
jumps.cpp:47:18: error: stray '`' in program
   47 |       - For each `c` from $1$ to $N$:
      |                  ^
jumps.cpp:47:20: error: stray '`' in program
   47 |       - For each `c` from $1$ to $N$:
      |                    ^
jumps.cpp:48:98: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                  ^
jumps.cpp:48:103: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                       ^
jumps.cpp:48:118: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                                      ^
jumps.cpp:48:119: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                                       ^
jumps.cpp:48:181: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                                                                                                     ^
jumps.cpp:48:188: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                                                                                                            ^
jumps.cpp:48:195: error: stray '\' in program
   48 |           - **Update the Fenwick Tree**: For a fixed $c$, we consider all pairs $(a, b)$ with $a \< b \< c$ and $a+c \\geq 2b$. The new value we can add to our data structure is $A\_a + A\_b + A\_c$.
      |                                                                                                                                                                                                   ^
jumps.cpp:49:112: error: stray '\' in program
   49 |               - This step is the most complex. The values we need to add to the Fenwick tree are of the form $A\_a + A\_b$.
      |                                                                                                                ^
jumps.cpp:49:119: error: stray '\' in program
   49 |               - This step is the most complex. The values we need to add to the Fenwick tree are of the form $A\_a + A\_b$.
      |                                                                                                                       ^
jumps.cpp:50:68: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                    ^
jumps.cpp:50:136: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                        ^
jumps.cpp:50:143: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                               ^
jumps.cpp:50:150: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                                      ^
jumps.cpp:50:183: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                                                                       ^
jumps.cpp:50:184: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                                                                        ^
jumps.cpp:50:206: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                                                                                              ^
jumps.cpp:50:207: error: stray '\' in program
   50 |               - For each $b$ from $1$ to $c-1$, we consider all $a \< b$. A new candidate for a triple jump is $(a, b, c)$, with sum $A\_a + A\_b + A\_c$. This is only valid if $a+c \\geq 2b$, which is $a \\geq 2b-c$.
      |                                                                                                                                                                                                               ^
jumps.cpp:51:26: error: stray '`' in program
   51 |               - For each `b`, we need to find the best `a`. This is $A\_b + RMQ(max(1, 2b-c), b-1)$.
      |                          ^
jumps.cpp:51:28: error: stray '`' in program
   51 |               - For each `b`, we need to find the best `a`. This is $A\_b + RMQ(max(1, 2b-c), b-1)$.
      |                            ^
jumps.cpp:51:56: error: stray '`' in program
   51 |               - For each `b`, we need to find the best `a`. This is $A\_b + RMQ(max(1, 2b-c), b-1)$.
      |                                                        ^
jumps.cpp:51:58: error: stray '`' in program
   51 |               - For each `b`, we need to find the best `a`. This is $A\_b + RMQ(max(1, 2b-c), b-1)$.
      |                                                          ^
jumps.cpp:51:71: error: stray '\' in program
   51 |               - For each `b`, we need to find the best `a`. This is $A\_b + RMQ(max(1, 2b-c), b-1)$.
      |                                                                       ^
jumps.cpp:53:105: error: stray '`' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                         ^
jumps.cpp:53:110: error: stray '`' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                              ^
jumps.cpp:53:144: error: stray '`' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                                                                ^
jumps.cpp:53:152: error: stray '`' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                                                                        ^
jumps.cpp:53:168: error: stray '`' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                                                                                        ^
jumps.cpp:53:173: error: stray '`' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                                                                                             ^
jumps.cpp:53:204: error: stray '\' in program
   53 |               - To optimize this, we can maintain the Fenwick tree values differently. The Fenwick tree `T[b]` will not directly store the sum `A_a+A_b`. Instead, let `T[k]` store the maximum value of $A\_a$ for a fixed $a$ and some optimal $b$.
      |                                                                                                                                                                                                            ^
jumps.cpp:54:84: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                    ^
jumps.cpp:54:88: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                        ^
jumps.cpp:54:95: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                               ^
jumps.cpp:54:97: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                 ^
jumps.cpp:54:104: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                        ^
jumps.cpp:54:108: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                            ^
jumps.cpp:54:166: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                      ^
jumps.cpp:54:168: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                        ^
jumps.cpp:54:181: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                                     ^
jumps.cpp:54:185: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                                         ^
jumps.cpp:54:195: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                                                   ^
jumps.cpp:54:197: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                                                     ^
jumps.cpp:54:205: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                                                             ^
jumps.cpp:54:234: error: stray '`' in program
   54 |               - A more refined approach is to use the Fenwick tree to maintain the `max` over `b`. Let `T_b` be a Fenwick tree over indices from $1$ to $N$. At step `c`, we update `T_b` for all `b` where `A_b + RMQ(max(1, 2b-c), b-1)` changes. This still looks slow.
      |                                                                                                                                                                                                                                          ^
jumps.cpp:56:4: warning: missing terminating ' character
   56 | Let's use a simpler, more direct Fenwick tree approach.
      |    ^
jumps.cpp:56:4: error: missing terminating ' character
   56 | Let's use a simpler, more direct Fenwick tree approach.
      |    ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
jumps.cpp:58:9: error: stray '`' in program
   58 |   - Let `T[i]` store the maximum value of $A\_a+A\_b+A\_c$ for a triple jump where the third take-off section is $i$, for all possible $a,b$.
      |         ^
jumps.cpp:58:14: error: stray '`' in program
   58 |   - Let `T[i]` store the maximum value of $A\_a+A\_b+A\_c$ for a triple jump where the third take-off section is $i$, for all possible $a,b$.
      |              ^
jumps.cpp:58:45: error: stray '\' in program
   58 |   - Let `T[i]` store the maximum value of $A\_a+A\_b+A\_c$ for a triple jump where the third take-off section is $i$, for all possible $a,b$.
      |                                             ^
jumps.cpp:58:50: error: stray '\' in program
   58 |   - Let `T[i]` store the maximum value of $A\_a+A\_b+A\_c$ for a triple jump where the third take-off section is $i$, for all possible $a,b$.
      |                                                  ^
jumps.cpp:58:55: error: stray '\' in program
   58 |   - Let `T[i]` store the maximum value of $A\_a+A\_b+A\_c$ for a triple jump where the third take-off section is $i$, for all possible $a,b$.
      |                                                       ^
jumps.cpp:59:13: error: stray '`' in program
   59 |   - Iterate `c` from `1` to `N`.
      |             ^
jumps.cpp:59:15: error: stray '`' in program
   59 |   - Iterate `c` from `1` to `N`.
      |               ^
jumps.cpp:59:22: error: stray '`' in program
   59 |   - Iterate `c` from `1` to `N`.
      |                      ^
jumps.cpp:59:24: error: stray '`' in program
   59 |   - Iterate `c` from `1` to `N`.
      |                        ^
jumps.cpp:59:29: error: stray '`' in program
   59 |   - Iterate `c` from `1` to `N`.
      |                             ^
jumps.cpp:59:31: error: stray '`' in program
   59 |   - Iterate `c` from `1` to `N`.
      |                               ^
jumps.cpp:60:13: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |             ^
jumps.cpp:60:15: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |               ^
jumps.cpp:60:30: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                              ^
jumps.cpp:60:34: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                  ^
jumps.cpp:60:72: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                        ^
jumps.cpp:60:79: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                               ^
jumps.cpp:60:86: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                                      ^
jumps.cpp:60:96: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                                                ^
jumps.cpp:60:102: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                                                      ^
jumps.cpp:60:112: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                                                                ^
jumps.cpp:60:125: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                                                                             ^
jumps.cpp:60:137: error: stray '`' in program
   60 |   - At step `c`, we consider `A_c` as the third section. For all pairs `(a, b)` with `a < b < c` and `a+c >= 2b`, the value `A_a+A_b+A_c` is a potential answer.
      |                                                                                                                                         ^
jumps.cpp:61:18: error: stray '`' in program
   61 |   - The value is `A_c + \max_{b<c, a<b, a+c \geq 2b} (A_a+A_b)`.
      |                  ^
jumps.cpp:61:25: error: stray '\' in program
   61 |   - The value is `A_c + \max_{b<c, a<b, a+c \geq 2b} (A_a+A_b)`.
      |                         ^
jumps.cpp:61:45: error: stray '\' in program
   61 |   - The value is `A_c + \max_{b<c, a<b, a+c \geq 2b} (A_a+A_b)`.
      |                                             ^
jumps.cpp:61:63: error: stray '`' in program
   61 |   - The value is `A_c + \max_{b<c, a<b, a+c \geq 2b} (A_a+A_b)`.
      |                                                               ^
jumps.cpp:62:9: error: stray '`' in program
   62 |   - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
      |         ^
jumps.cpp:62:10: error: stray '\' in program
   62 |   - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
      |          ^
jumps.cpp:62:14: error: stray '`' in program
   62 |   - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
      |              ^
jumps.cpp:62:69: error: stray '`' in program
   62 |   - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
      |                                                                     ^
jumps.cpp:62:73: error: stray '`' in program
   62 |   - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
      |                                                                         ^
jumps.cpp:62:104: error: stray '`' in program
   62 |   - The `\max` part can be computed using another Fenwick tree. Let `F_c` be a Fenwick tree on indices `b` from $1$ to $N$.
      |