#include "migrations.h"
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <climits>
#include <cmath>
#include <complex>
#include <cstring>
#include <functional>
#include <iomanip>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <vector>
using namespace std;
using ll = long long;
using db = long double; // or double, if TL is tight
using str = string; // yay python!
// pairs
using pi = pair<int, int>;
using pl = pair<ll, ll>;
using pd = pair<db, db>;
#define mp make_pair
#define f first
#define s second
#define tcT template <class T
#define tcTU tcT, class U
// ^ lol this makes everything look weird but I'll try it
tcT > using V = vector<T>;
tcT, size_t SZ > using AR = array<T, SZ>;
using vi = V<int>;
using vb = V<bool>;
using vl = V<ll>;
using vd = V<db>;
using vs = V<str>;
using vpi = V<pi>;
using vpl = V<pl>;
using vpd = V<pd>;
// vectors
#define sz(x) int(size(x))
#define bg(x) begin(x)
#define all(x) bg(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define sor(x) sort(all(x))
#define rsz resize
#define ins insert
#define pb push_back
#define eb emplace_back
#define ft front()
#define bk back()
#define lb lower_bound
#define ub upper_bound
tcT > int lwb(const V<T> &a, const T &b) { return int(lb(all(a), b) - bg(a)); }
tcT > int upb(const V<T> &a, const T &b) { return int(ub(all(a), b) - bg(a)); }
// loops
#define FOR(i, a, b) for (int i = (a); i < (b); ++i)
#define F0R(i, a) FOR(i, 0, a)
#define ROF(i, a, b) for (int i = (b) - 1; i >= (a); --i)
#define R0F(i, a) ROF(i, 0, a)
#define rep(a) F0R(_, a)
#define each(a, x) for (auto &a : x)
const int MOD = 998244353; // 1e9+7;
const int MX = (int)2e5 + 5;
const ll BIG = 1e18; // not too close to LLONG_MAX
const db PI = acos((db)-1);
const int dx[4]{1, 0, -1, 0}, dy[4]{0, 1, 0, -1}; // for every grid problem!!
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count());
template <class T> using pqg = priority_queue<T, vector<T>, greater<T>>;
// bitwise ops
// also see https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html
constexpr int pct(int x) { return __builtin_popcount(x); } // # of bits set
constexpr int bits(int x) { // assert(x >= 0); // make C++11 compatible until
// USACO updates ...
return x == 0 ? 0 : 31 - __builtin_clz(x);
} // floor(log2(x))
constexpr int p2(int x) { return 1 << x; }
constexpr int msk2(int x) { return p2(x) - 1; }
ll cdiv(ll a, ll b) {
return a / b + ((a ^ b) > 0 && a % b);
} // divide a by b rounded up
ll fdiv(ll a, ll b) {
return a / b - ((a ^ b) < 0 && a % b);
} // divide a by b rounded down
tcT > bool ckmin(T &a, const T &b) {
return b < a ? a = b, 1 : 0;
} // set a = min(a,b)
tcT > bool ckmax(T &a, const T &b) {
return a < b ? a = b, 1 : 0;
} // set a = max(a,b)
tcTU > T fstTrue(T lo, T hi, U f) {
++hi;
assert(lo <= hi); // assuming f is increasing
while (lo < hi) { // find first index such that f is true
T mid = lo + (hi - lo) / 2;
f(mid) ? hi = mid : lo = mid + 1;
}
return lo;
}
tcTU > T lstTrue(T lo, T hi, U f) {
--lo;
assert(lo <= hi); // assuming f is decreasing
while (lo < hi) { // find first index such that f is true
T mid = lo + (hi - lo + 1) / 2;
f(mid) ? lo = mid : hi = mid - 1;
}
return lo;
}
tcT > void remDup(vector<T> &v) { // sort and remove duplicates
sort(all(v));
v.erase(unique(all(v)), end(v));
}
tcTU > void safeErase(T &t, const U &u) {
auto it = t.find(u);
assert(it != end(t));
t.erase(it);
}
inline namespace IO {
#define SFINAE(x, ...) \
template <class, class = void> struct x : std::false_type {}; \
template <class T> struct x<T, std::void_t<__VA_ARGS__>> : std::true_type {}
SFINAE(DefaultI, decltype(std::cin >> std::declval<T &>()));
SFINAE(DefaultO, decltype(std::cout << std::declval<T &>()));
SFINAE(IsTuple, typename std::tuple_size<T>::type);
SFINAE(Iterable, decltype(std::begin(std::declval<T>())));
template <auto &is> struct Reader {
template <class T> void Impl(T &t) {
if constexpr (DefaultI<T>::value) is >> t;
else if constexpr (Iterable<T>::value) {
for (auto &x : t) Impl(x);
} else if constexpr (IsTuple<T>::value) {
std::apply([this](auto &...args) { (Impl(args), ...); }, t);
} else static_assert(IsTuple<T>::value, "No matching type for read");
}
template <class... Ts> void read(Ts &...ts) { ((Impl(ts)), ...); }
};
template <class... Ts> void re(Ts &...ts) { Reader<cin>{}.read(ts...); }
#define def(t, args...) \
t args; \
re(args);
template <auto &os, bool debug, bool print_nd> struct Writer {
string comma() const { return debug ? "," : ""; }
template <class T> constexpr char Space(const T &) const {
return print_nd && (Iterable<T>::value or IsTuple<T>::value) ? '\n'
: ' ';
}
template <class T> void Impl(T const &t) const {
if constexpr (DefaultO<T>::value) os << t;
else if constexpr (Iterable<T>::value) {
if (debug) os << '{';
int i = 0;
for (auto &&x : t)
((i++) ? (os << comma() << Space(x), Impl(x)) : Impl(x));
if (debug) os << '}';
} else if constexpr (IsTuple<T>::value) {
if (debug) os << '(';
std::apply(
[this](auto const &...args) {
int i = 0;
(((i++) ? (os << comma() << " ", Impl(args)) : Impl(args)),
...);
},
t);
if (debug) os << ')';
} else static_assert(IsTuple<T>::value, "No matching type for print");
}
template <class T> void ImplWrapper(T const &t) const {
if (debug) os << "\033[0;31m";
Impl(t);
if (debug) os << "\033[0m";
}
template <class... Ts> void print(Ts const &...ts) const {
((Impl(ts)), ...);
}
template <class F, class... Ts>
void print_with_sep(const std::string &sep, F const &f,
Ts const &...ts) const {
ImplWrapper(f), ((os << sep, ImplWrapper(ts)), ...), os << '\n';
}
void print_with_sep(const std::string &) const { os << '\n'; }
};
template <class... Ts> void pr(Ts const &...ts) {
Writer<cout, false, true>{}.print(ts...);
}
template <class... Ts> void ps(Ts const &...ts) {
Writer<cout, false, true>{}.print_with_sep(" ", ts...);
}
} // namespace IO
inline namespace Debug {
template <typename... Args> void err(Args... args) {
Writer<cerr, true, false>{}.print_with_sep(" | ", args...);
}
template <typename... Args> void errn(Args... args) {
Writer<cerr, true, true>{}.print_with_sep(" | ", args...);
}
void err_prefix(str func, int line, string args) {
cerr << "\033[0;31m\u001b[1mDEBUG\033[0m"
<< " | "
<< "\u001b[34m" << func << "\033[0m"
<< ":"
<< "\u001b[34m" << line << "\033[0m"
<< " - "
<< "[" << args << "] = ";
}
#ifdef LOCAL
#define dbg(args...) err_prefix(__FUNCTION__, __LINE__, #args), err(args)
#define dbgn(args...) err_prefix(__FUNCTION__, __LINE__, #args), errn(args)
#else
#define dbg(...)
#define dbgn(args...)
#endif
const auto beg_time = std::chrono::high_resolution_clock::now();
// https://stackoverflow.com/questions/47980498/accurate-c-c-clock-on-a-multi-core-processor-with-auto-overclock?noredirect=1&lq=1
double time_elapsed() {
return chrono::duration<double>(std::chrono::high_resolution_clock::now() -
beg_time)
.count();
}
} // namespace Debug
inline namespace FileIO {
void setIn(str s) { freopen(s.c_str(), "r", stdin); }
void setOut(str s) { freopen(s.c_str(), "w", stdout); }
void setIO(str s = "") {
cin.tie(0)->sync_with_stdio(0); // unsync C / C++ I/O streams
cout << fixed << setprecision(12);
// cin.exceptions(cin.failbit);
// throws exception when do smth illegal
// ex. try to read letter into int
if (sz(s)) setIn(s + ".in"), setOut(s + ".out"); // for old USACO
}
} // namespace FileIO
const int D = 14;
V<AR<int, 14>> p;
vi depth;
AR<int, 2> diameter;
AR<vi, 2> cands;
int lca(int a, int b) {
if (depth.at(a) < depth.at(b)) swap(a, b);
R0F(d, D) if ((depth.at(a) - depth.at(b)) & (1 << d)) a = p.at(a).at(d);
if (a == b) return a;
R0F(d, D) if (p.at(a).at(d) != p.at(b).at(d)) {
a = p.at(a).at(d);
b = p.at(b).at(d);
}
return p.at(a).at(0);
}
int dist(int a, int b) {
return depth.at(a) + depth.at(b) - 2 * depth.at(lca(a, b));
}
int getpot(pi p, int cons = 1) { return (p.f + cons) * (p.s + cons); }
template <class T> V<T> extract(V<T> v, int parts, int ret) {
assert(sz(v));
while (sz(v) % parts) v.pb(v.bk);
assert(0 <= ret && ret < parts);
return V<T>(begin(v) + sz(v) / parts * ret,
begin(v) + sz(v) / parts * (ret + 1));
}
int idx(vi &v, int goal, int parts) {
dbg("IDX", sz(v), v.ft, goal, parts);
assert(sz(v));
while (sz(v) % parts) v.pb(v.bk);
int i = 0;
while (v.at(i) != goal) {
++i;
assert(i < sz(v));
}
int ret = i * parts / sz(v);
v = extract(v, parts, ret);
dbg("RET", ret);
return ret;
}
// 10000, 182 (1 move)
// 553, 25 (1 move)
// 81, 6 (1 move)
// 23, rest (6 moves)
vi cuts;
void init_cuts(int N) {
cuts = {N}; // cuts[6]
cuts.pb(cuts.bk - 3); // cuts[5]
cuts.pb(cuts.bk - 2); // cuts[4]
cuts.pb(cuts.bk - 2); // cuts[3]
cuts.pb(cuts.bk - 6); // cuts[2]
cuts.pb(cuts.bk - 20); // cuts[1]
cuts.pb(cuts.bk - 140); // cuts[0]
reverse(all(cuts));
assert(sz(cuts) == 7);
}
pi pos_val;
// full credit
vi with_label;
const V<V<AR<int, 2>>> partitions{{{0, 4}, {0, 5}, {0, 6}, {0, 8}, {8, 4}},
{{0, 7}, {1, 7}, {1, 4}, {1, 8}, {8, 7}},
{{1, 5}, {1, 6}, {2, 6}, {2, 8}, {8, 6}},
{{2, 4}, {2, 5}, {3, 5}, {3, 8}, {8, 5}},
{{3, 4}, {3, 6}, {3, 7}, {2, 7}}};
V<AR<int, 2>> cand_pairs;
bool common(AR<int, 2> a, AR<int, 2> b) {
each(x, a) each(y, b) if (x == y) return true;
return false;
}
V<AR<int, 2>> add_all(V<AR<int, 2>> v, int nvert) {
V<AR<int, 2>> cands = v;
each(t, v) {
cands.pb({t[0], nvert});
cands.pb({t[1], nvert});
}
remDup(cands);
return cands;
}
V<AR<int, 2>> valid_ordering(V<AR<int, 2>> v) {
sor(v);
do {
bool ok = true;
for (int i = 0; i + 1 < sz(v); i += 2) {
if (!common(v[i], v[i + 1])) {
ok = false;
break;
}
}
if (ok) return v;
} while (next_permutation(all(v)));
assert(false);
}
AR<int, 2> pair_to_edge(AR<int, 2> t) {
return {with_label.at(t[0]), with_label.at(t[1])};
}
bool edge_eq(AR<int, 2> a, AR<int, 2> b) {
if (a == b) return true;
if (a == AR<int, 2>{b[1], b[0]}) return true;
return false;
}
int c2(int x) { return x * (x + 1) / 2; };
int send_message(int N, int i, int Pi) {
if (i == 1) {
p.eb(), depth.eb();
F0R(j, 2) cands.at(j).pb(0);
init_cuts(N);
}
p.eb();
p.bk.at(0) = Pi;
depth.pb(depth.at(Pi) + 1);
FOR(d, 1, D) p.bk.at(d) = p.at(p.bk.at(d - 1)).at(d - 1);
assert(sz(p) == i + 1);
F0R(j, 2) {
if (dist(i, diameter.at(j)) > dist(diameter.at(0), diameter.at(1))) {
diameter.at(j ^ 1) = i;
break;
}
}
int cut_idx = 0;
while (cuts.at(cut_idx + 1) <= i) ++cut_idx;
if (cut_idx > 0 && cuts.at(cut_idx) == i) {
FOR(j, cuts.at(cut_idx - 1) + 1, cuts.at(cut_idx)) {
cands.at(0).pb(j);
cands.at(1).pb(j);
}
}
int msg = 0;
if (i >= cuts.at(5)) {
if (i == cuts.at(5)) {
dbg("e phase 2", i, sz(cands.at(0)));
assert(sz(cands.at(0)) == 4 && sz(cands.at(1)) == 4);
with_label = {};
with_label.ins(end(with_label), all(cands.at(0)));
with_label.ins(end(with_label), all(cands.at(1)));
with_label.pb(i);
while (true) {
bool found = 0;
each(t, partitions.at(msg)) {
if (diameter == pair_to_edge(t)) { found = true; }
}
if (found) break;
++msg;
}
cand_pairs = partitions.at(msg);
} else if (i == cuts.at(5) + 1) {
dbg("e phase 3", cand_pairs);
with_label.pb(i);
cand_pairs = add_all(cand_pairs, 9);
cand_pairs = valid_ordering(cand_pairs);
while (sz(cand_pairs) < 10) cand_pairs.pb(cand_pairs.bk);
assert(sz(cand_pairs) == 10);
int ret = 0;
while (!edge_eq(diameter, pair_to_edge(cand_pairs.at(ret)))) {
++ret;
}
msg = ret / 2;
cand_pairs = extract(cand_pairs, 5, msg);
} else {
dbg("e phase 4", cand_pairs);
with_label.pb(i);
assert(i == cuts.at(5) + 2);
cand_pairs = add_all(cand_pairs, 10);
assert(sz(cand_pairs) <= 5);
while (!edge_eq(diameter, pair_to_edge(cand_pairs.at(msg)))) ++msg;
dbg(diameter);
}
} else if (i >= cuts.at(0)) {
if (i == cuts.at(cut_idx)) {
cands.at(0).pb(i);
cands.at(1).pb(i);
dbg("phase 1", i, sz(cands.at(0)), sz(cands.at(1)));
assert(sz(cands.at(0)) == sz(cands.at(1)));
const int space = cuts.at(cut_idx + 1) - cuts.at(cut_idx);
const int num_cands = 4 * space + 1;
int to_encrypt = 0;
if (cut_idx == 0) {
int sqrt_cands = 1;
while (c2(sqrt_cands + 1) <= num_cands) ++sqrt_cands;
if (diameter.at(0) < diameter.at(1))
swap(diameter.at(0), diameter.at(1));
to_encrypt = c2(idx(cands.at(0), diameter.at(0), sqrt_cands)) +
idx(cands.at(1), diameter.at(1), sqrt_cands);
} else {
int sqrt_cands = sqrt(num_cands);
F0R(j, 2) {
to_encrypt = sqrt_cands * to_encrypt +
idx(cands.at(j), diameter.at(j), sqrt_cands);
}
}
assert(0 <= to_encrypt && to_encrypt < num_cands);
dbg(i, to_encrypt);
if (to_encrypt == 4 * space) {
pos_val = {-1, -1};
} else {
pos_val = {i + to_encrypt / 4, 1 + to_encrypt % 4};
}
}
if (i == pos_val.f) {
assert(1 <= pos_val.s && pos_val.s <= 4);
msg = pos_val.s;
}
} else {
F0R(j, 2) cands.at(j).pb(i);
}
return msg;
}
pi arr_to_pair(AR<int, 2> a) { return {a[0], a[1]}; }
std::pair<int, int> longest_path(std::vector<int> S) {
p = {};
depth = {};
diameter = {};
cands = {};
int N = sz(S);
init_cuts(N);
F0R(i, N) {
// dbg("AT", i);
int cut_idx = 0;
while (cuts.at(cut_idx + 1) <= i) ++cut_idx;
if (cut_idx > 0 && cuts.at(cut_idx) == i) {
int space = cuts.at(cut_idx) - cuts.at(cut_idx - 1);
int num_cands = 4 * space + 1;
int sqrt_cands = 0;
if (cut_idx == 1) {
while (c2(sqrt_cands + 1) <= num_cands) ++sqrt_cands;
} else {
sqrt_cands = sqrt(num_cands);
}
int to_encrypt = 0;
if (pos_val == mp(-1, -1)) to_encrypt = 4 * space;
else {
to_encrypt =
(pos_val.f - cuts.at(cut_idx - 1)) * 4 + (pos_val.s - 1);
}
assert(0 <= to_encrypt && to_encrypt < num_cands);
AR<int, 2> idx{};
if (cut_idx == 1) {
while (c2(idx.at(0) + 1) <= to_encrypt) ++idx.at(0);
idx.at(1) = to_encrypt - c2(idx.at(0));
} else {
idx = {to_encrypt / sqrt_cands, to_encrypt % sqrt_cands};
}
dbg("rextract", cuts.at(cut_idx - 1), pos_val, to_encrypt, cut_idx,
sqrt_cands, idx);
F0R(j, 2)
cands.at(j) = extract(cands.at(j), sqrt_cands, idx.at(j));
FOR(j, cuts.at(cut_idx - 1) + 1, cuts.at(cut_idx)) {
cands.at(0).pb(j);
cands.at(1).pb(j);
}
dbg("DONE");
}
int msg = S.at(i);
if (i >= cuts.at(5)) {
if (i == cuts.at(5)) {
dbg("d phase 2", i, sz(cands.at(0)));
dbg(cands.at(0), cands.at(1));
assert(sz(cands.at(0)) == 4 && sz(cands.at(1)) == 4);
with_label = {};
with_label.ins(end(with_label), all(cands.at(0)));
with_label.ins(end(with_label), all(cands.at(1)));
with_label.pb(i);
cand_pairs = partitions.at(msg);
} else if (i == cuts.at(5) + 1) {
dbg("d phase 3", i, cand_pairs);
with_label.pb(i);
cand_pairs = add_all(cand_pairs, 9);
cand_pairs = valid_ordering(cand_pairs);
while (sz(cand_pairs) < 10) cand_pairs.pb(cand_pairs.bk);
assert(sz(cand_pairs) == 10);
cand_pairs = extract(cand_pairs, 5, msg);
} else {
dbg("d phase 4", i, cand_pairs);
assert(i == cuts.at(5) + 2);
with_label.pb(i);
assert(i == cuts.at(5) + 2);
cand_pairs = add_all(cand_pairs, 10);
assert(sz(cand_pairs) <= 5);
return arr_to_pair(pair_to_edge(cand_pairs.at(msg)));
}
} else if (i >= cuts.at(0)) {
if (i == cuts.at(cut_idx)) {
pos_val = {-1, -1};
cands.at(0).pb(i);
cands.at(1).pb(i);
}
if (msg != 0) {
assert(pos_val == mp(-1, -1));
pos_val = {i, msg};
}
} else {
assert(msg == 0);
F0R(j, 2) cands.at(j).pb(i);
}
if (i == N - 1) {
F0R(j, 2) assert(sz(cands.at(j)) == 1);
return {cands.at(0).ft, cands.at(1).ft};
}
}
assert(false);
}
