#include <bits/stdc++.h>
#include "gondola.h"
using namespace std;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pi;
typedef pair<ll, ll> pl;
typedef vector<int> vi;
typedef vector<ld> vd;
typedef vector<ll> vl;
#define pb push_back
#define f first
#define s second
namespace debug {
const int DEBUG = true;
template<class T1, class T2>
void pr(const pair<T1, T2> &x);
template<class T, size_t SZ>
void pr(const array<T, SZ> &x);
template<class T>
void pr(const vector<T> &x);
template<class T>
void pr(const set<T> &x);
template<class T1, class T2>
void pr(const map<T1, T2> &x);
template<class T>
void pr(const T &x) { if (DEBUG) cout << x; }
template<class T, class... Ts>
void pr(const T &first, const Ts &... rest) { pr(first), pr(rest...); }
template<class T1, class T2>
void pr(const pair<T1, T2> &x) { pr("{", x.f, ", ", x.s, "}"); }
template<class T>
void prIn(const T &x) {
pr("{");
bool fst = 1;
for (auto &a : x) {
pr(fst ? "" : ", ", a), fst = 0;
}
pr("}");
}
template<class T, size_t SZ>
void pr(const array<T, SZ> &x) { prIn(x); }
template<class T>
void pr(const vector<T> &x) { prIn(x); }
template<class T>
void pr(const set<T> &x) { prIn(x); }
template<class T1, class T2>
void pr(const map<T1, T2> &x) { prIn(x); }
void ps() { pr("\n"), cout << flush; }
template<class Arg, class... Args>
void ps(const Arg &first, const Args &... rest) {
pr(first, " ");
ps(rest...);
}
}
using namespace debug;
int valid(int N, int a[]) {
set<int> seen;
int off = -1;
for (int i = 0; i < N; i++) {
if (seen.count(a[i])) {
return false;
}
seen.insert(a[i]);
if (a[i] <= N) {
int nOff = (N + i - (a[i] - 1)) % N;
if (off == -1) {
off = nOff;
} else {
if (off != nOff) {
return false;
}
}
}
}
return true;
}
int replacement(int N, int a[], int rep[]) {
int off = 0;
for (int i = 0; i < N; i++) {
if (a[i] <= N) {
off = (N + i - (a[i] - 1)) % N;
}
}
// ps(off);
vi mod(a, a + N);
for (int i = 0; i < N; i++) {
if (mod[i] > N) {
mod[i] = (i + 1 + N - off) % N;
if (mod[i] == 0) {
mod[i] = N;
}
}
}
// ps(mod);
vector<pi> ord;
for (int i = 0; i < N; i++) {
if (a[i] > N) {
ord.pb({a[i], i});
}
}
sort(ord.begin(), ord.end());
int nxt = N + 1;
int L = 0;
for (pi cur : ord) {
while (nxt <= cur.f) {
rep[L++] = mod[cur.s];
mod[cur.s] = nxt++;
}
}
return L;
}
const ll MOD = 1e9 + 9;
ll binExp(ll b, ll p) {
// ps(b, p);
b %= MOD;
ll c = 1;
while (p > 0) {
if (p & 1) {
c = c * b % MOD;
}
b = b * b % MOD;
p >>= 1;
}
return c;
}
int countReplacement(int N, int a[]) {
if (!valid(N, a)) {
return 0;
}
int fixed = 0;
vi free;
for (int i = 0; i < N; i++) {
if (a[i] <= N) {
fixed++;
} else {
free.pb(a[i]);
}
}
// ps(fixed);
// ps(free);
sort(free.begin(), free.end());
int nxt = N + 1;
ll cnt = 1;
for (int i = 0; i < free.size(); i++) {
cnt = cnt * binExp(N - fixed - i, free[i] - nxt) % MOD;
nxt = free[i] + 1;
}
if (fixed == 0) {
cnt = cnt * N % MOD;
}
return cnt;
}
//int main() {
// int inp[] = {3, 4};
// ps(countReplacement(2, inp));
//}
