#include "gondola.h"
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#define eb emplace_back
#define pb push_back
#define ft first
#define sd second
#define pi pair<int, int>
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define dbg(...) dbg_out(__VA_ARGS__)
using ll = long long;
using ld = long double;
using namespace std;
using namespace __gnu_pbds;
//Constants
const ll INF = 5 * 1e18;
const int IINF = 2 * 1e9;
const ll MOD = 1e9 + 9;
// const ll MOD = 998244353;
const ll dx[4] = {1, 0, -1, 0}, dy[4] = {0, 1, 0, -1};
const ld PI = 3.14159265359;
//Templates
template<typename A, typename B> ostream& operator<<(ostream &os, const pair<A, B> &p) {return os << '(' << p.first << ", " << p.second << ')';}
template<typename T_container, typename T = typename enable_if<!is_same<T_container, string>::value, typename T_container::value_type>::type> ostream& operator<<(ostream &os, const T_container &v) {os << '['; string sep; for (const T &x : v) os << sep << x, sep = ", "; return os << ']';}
void dbg_out() {cerr << endl;}
template<typename Head, typename... Tail> void dbg_out(Head H, Tail... T) { cerr << H << ' '; dbg_out(T...); }
template<typename T> void mins(T& x, T y) {x = min(x, y);}
template<typename T> void maxs(T& x, T y) {x = max(x, y);}
template<typename T> using oset = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
template<typename T> using omset = tree<T, null_type, less_equal<T>, rb_tree_tag, tree_order_statistics_node_update>;
int valid(int n, int inputSeq[]) {
set<int> st;
int mini = 0;
for (int i = 0; i < n; i++) {
st.insert(inputSeq[i]);
if (inputSeq[i] < inputSeq[mini])
mini = i;
}
if (n != (int) st.size()) return 0;
if (inputSeq[mini] > n) return 1;
for (int i = 0; i < n; i++) {
int pos = (mini + i) % n;
if (inputSeq[pos] > n) continue;
if (inputSeq[pos] != inputSeq[mini] + i) return 0;
}
return 1;
}
int replacement(int n, int gondolaSeq[], int replacementSeq[]) {
int mini = 0;
for (int i = 0; i < n; i++) {
if (gondolaSeq[i] < gondolaSeq[mini])
mini = i;
}
int val = gondolaSeq[mini];
if (val > n) mini = 0, val = 1;
vector<pi> ar(n);
for (int i = 0; i < n; i++) {
int pos = (mini + i) % n;
ar[pos] = {gondolaSeq[pos], val};
val++; if (val > n) val = 1;
}
sort(all(ar));
int pp = 0;
int z = n + 1;
for (int i = 0; i < n; i++) {
if (ar[i].ft <= n) continue;
int cur = ar[i].sd;
while (cur < ar[i].ft) {
replacementSeq[pp++] = cur;
cur = z++;
}
}
return pp;
}
ll bp(ll a, ll n) {
if (n == 0) return 1ll;
if (n & 1) return a * bp(a, n - 1) % MOD;
ll h = bp(a, n / 2);
return h * h % MOD;
}
int countReplacement(int n, int inputSeq[]) {
if (!valid(n, inputSeq)) return 0;
int bad = 0;
for (int i = 0; i < n; i++) {
if (inputSeq[i] > n) bad++;
}
sort(inputSeq, inputSeq + n);
ll ans = (bad == n ? n : 1); int cur = n + 1;
for (int i = 0; i < n; i++) {
if (inputSeq[i] <= n) continue;
int k = inputSeq[i] - cur;
(ans *= bp(bad, k)) %= MOD;
cur = inputSeq[i] + 1;
bad--;
}
return ans;
}
