#include <bits/stdc++.h>
#include "gondola.h"
using namespace std;
using ll = long long;
using pii = pair<int, int>;
#define pb push_back
#define ff first
#define ss second
const int mod = 1e9 + 9;
int valid(int n, int x[]){
map<int, bool> f;
int m = 0;
for (int i = 0; i < n; i++){
if (f.find(x[i]) != f.end()) return 0;
f[x[i]] = 1;
m = max(m, x[i]);
}
pii mn = {1e9, 0};
for (int i = 0; i < n; i++) mn = min(mn, {x[i], i});
int p = mn.ss;
vector<int> y(n);
for (int i = 0; i < n; i++){
int j = i - p;
if (j < 0) j += n;
y[j] = x[i];
}
int X = y[0];
for (int i = 1; i < n; i++){
if (y[i] > n) continue;
int Y = y[i] - i;
if (Y < 0) Y += n;
if (Y != X) return 0;
}
if (m >= 2 * n) return 1;
int cc = 0;
for (int i = 1; i <= n; i++) cc += f[i];
if ((n - cc) > (m - n)) return 0;
return 1;
}
int replacement(int n, int a[], int b[]){
int m = 0;
for (int i = 0; i < n; i++) m = max(m, a[i]);
int p = 0;
for (int i = 0; i < n; i++){
if (a[i] <= n){
p = i;
}
}
p = (a[p] - p - 1) % n;
if (p < 0) p += n;
vector<int> x(n);
for (int i = 0; i < n; i++){
int j = i + p;
if (j >= n) j -= n;
x[j] = a[i];
}
vector<pii> all;
for (int i = 0; i < n; i++){
if (x[i] > n) all.pb({x[i], i});
}
vector<int> f(n);
for (int i = 0; i < n; i++) f[i] = i + 1;
sort(all.begin(), all.end());
int t = 0;
for (auto [v, k]: all){
while (n < v){
b[t++] = f[k];
f[k] = ++n;
}
}
return t;
}
int bin_pow(int x, int n){
if (!n) return 1;
int out = x, pr = x; n--;
while (n > 0){
if (n & 1){
out = (1LL * out * pr) % mod;
}
pr = (1LL * pr * pr) % mod;
n >>= 1;
}
return out;
}
int countReplacement(int n, int a[]){
if (!valid(n, a)) return 0;
vector<int> all;
for (int i = 0; i < n; i++){
if (a[i] > n){
all.pb(a[i]);
}
}
sort(all.begin(), all.end());
int pre = n, out = 1;
for (int i = 0; i < all.size(); i++){
out = (1LL * out * bin_pow((int) all.size() - i, all[i] - pre - 1)) % mod;
pre = all[i];
}
if (all.size() == n){
out = (1LL * out * n) % mod;
}
return out;
}
