#include "gondola.h"
#include <bits/stdc++.h>
using namespace std;
#define vt vector
#define size(x) (int((x).size()))
#define all(x) begin(x), end(x)
#define REP(a, b, c, d) for (int a = (b); (d) > 0 ? a <= (c) : a >= (c); a += (d))
#define FOR(a, b, c) REP(a, b, c, 1)
#define ROF(a, b, c) REP(a, b, c, -1)
int valid(int N, int *A) {
map<int, int> cnt;
FOR(i, 0, N-1)
if (++cnt[--A[i]] > 1)
return 0;
FOR(i, 0, N-1)
if (A[i] < N) {
FOR(j, 0, N-1)
if (A[(i+j)%N] < N && A[(i+j)%N] != (A[i]+j)%N)
return 0;
break;
}
return 1;
}
int replacement(int N, int *A, int *ans) {
int start = 0;
FOR(i, 0, N-1)
if (A[i]-- < 2)
start = i;
vt<int> B(N);
FOR(i, 0, N-1)
B[(start+i)%N] = i;
vt<int> ord(N);
iota(all(ord), 0);
sort(all(ord), [&](const int &a, const int &b) {
return A[a] < A[b];
});
vt<int> res;
FOR(ii, 0, N-1) {
int i = ord[ii];
if (A[i] < N)
continue;
res.push_back(B[i]);
B[i] = N + size(res) - 1;
for (; B[i] < A[i]; B[i]++)
res.push_back(B[i]);
}
FOR(i, 0, size(res)-1)
ans[i] = res[i] + 1;
return size(res);
}
constexpr int MOD = 1e9 + 9;
int binpow(int a, int b) {
int ret = 1;
for (; b; b >>= 1, a = 1ll * a * a % MOD)
if (b & 1)
ret = 1ll * ret * a % MOD;
return ret;
}
int countReplacement(int N, int *A) {
if (!valid(N, A))
return 0;
sort(A, A+N);
int ans = 1;
FOR(i, 0, N-1)
if (A[i] >= N) {
#ifdef DEBUG
cout << "bruh: " << binpow(N - i, A[i] - (i ? max(N-1, A[i-1])+1 : N)) << ' ' << A[i] << ' ' << A[i-1] << '\n';
#endif
ans = 1ll * ans * binpow(N - i, A[i] - (i ? max(N-1, A[i-1])+1 : N)) % MOD;
}
return 1ll * ans * (A[0] >= N ? N : 1) % MOD;
}
