#include "gondola.h"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using i64 = long long;
void __print(int x) { cerr << x; } void __print(long x) { cerr << x; } void __print(long long x) { cerr << x; } void __print(unsigned x) { cerr << x; } void __print(unsigned long x) { cerr << x; } void __print(unsigned long long x) { cerr << x; } void __print(float x) { cerr << x; } void __print(double x) { cerr << x; } void __print(long double x) { cerr << x; } void __print(char x) { cerr << '\'' << x << '\''; } void __print(const char* x) { cerr << '"' << x << '"'; } void __print(const string& x) { cerr << '"' << x << '"'; } void __print(bool x) { cerr << (x ? "true" : "false"); } template<typename T, typename V> void __print(const pair<T, V>& x) { cerr << '{'; __print(x.first); cerr << ','; __print(x.second); cerr << '}'; } template<typename T> void __print(const T& x) { int f = 0; cerr << '{'; for (auto& i : x) cerr << (f++ ? "," : ""), __print(i); cerr << "}"; } void _print() { cerr << "]\n"; } template <typename T, typename... V> void _print(T t, V... v) { __print(t); if (sizeof...(v)) cerr << ", "; _print(v...); }
#define dbg(x...) cerr << "[" << #x << "] = ["; _print(x)
#define ln "\n"
#define fastIO() ios_base::sync_with_stdio(false); cin.tie(0); cout.tie(0)
#define all(x) (x).begin(), (x).end()
#define rep(i,a,b) for(int i=a; i<=b; ++i)
#define ar array
int Bit(int mask, int b) { return (mask >> b) & 1; }
const ll base = 311, MOD = 998244353, M = 1e9 + 9, INF = 1e18;
int valid(int n, int inputSeq[]) {
rep(i, 0, n - 1) inputSeq[i]--;
int x = -1;
rep(i, 0, n - 1)
if (inputSeq[i] < n) {
x = (inputSeq[i] - i + n) % n;
break;
}
map<int, int> mp;
rep(i, 0, n - 1) {
mp[inputSeq[i]]++;
if (mp[inputSeq[i]] > 1) return 0;
}
if (x == -1)
return 1;
rep(i, 0, n - 1)
if (inputSeq[i] < n &&
x != (inputSeq[i] - i + n) % n)
return 0;
return 1;
}
//----------------------
int replacement(int n, int gondolaSeq[], int replacementSeq[]) {
vector<int> a(n);
rep(i, 0, n - 1)
a[i] = gondolaSeq[i] - 1;
int x = 0;
rep(i, 0, n - 1) if (a[i] < n)
x = (a[i] - i + n) % n;
vector<ar<int, 2>> f(n);
rep(i, 0, n - 1)
f[i] = { a[i], (i + x) % n };
sort(all(f));
int l = 0, cur = n - 1;
rep(i, 0, n - 1) {
if (f[i][0] >= n) {
rep(j, cur + 1, f[i][0]) {
replacementSeq[l++] = f[i][1] + 1;
f[i][1] = j;
}
cur = f[i][0];
}
}
return l;
}
//----------------------
ll power(ll x, ll n) {
ll res = 1;
for (; n > 0; n >>= 1) {
if (n & 1)
res = (res * x) % M;
x = (x * x) % M;
}
return res;
}
int countReplacement(int n, int inputSeq[]) {
if (!valid(n, inputSeq)) return 0;
vector<int> a(n);
rep(i, 0, n - 1)
a[i] = inputSeq[i];
int x = -1;
rep(i, 0, n - 1) if (a[i] < n)
x = (a[i] - i + n) % n;
ll res = 1;
if (x == -1) {
res = n;
x = 0;
}
vector<ar<int, 2>> f(n);
rep(i, 0, n - 1)
f[i] = { a[i], (i + x) % n };
sort(all(f));
int cur = n - 1;
ll count = 0;
rep(i, 0, n - 1)
count += (f[i][0] > f[i][1]);
rep(i, 0, n - 1) {
if (f[i][0] >= n) {
// res *= 'count (f[i][1] < f[i][0])' ^ (f[i][0]-cur-1)
// rep(j, cur + 1, f[i][0]) {
// replacementSeq[l++] = f[i][1] + 1;
// f[i][1] = j;
// }
res = (res * power(count, f[i][0] - cur - 1)) % M;
f[i][1] = f[i][0];
--count;
cur = f[i][0];
}
}
return res;
}
// int main() {
// return 0;
// }
