#include <bits/stdc++.h>
#include "gondola.h"
using namespace std;
#define f first
#define s second
#define pb push_back
#define ar array
#define all(x) x.begin(), x.end()
#define siz(x) (int)x.size()
#define FOR(x, y, z) for(int x = (y); x < (z); x++)
#define ROF(x, z, y) for(int x = (y-1); x >= (z); x--)
#define F0R(x, z) FOR(x, 0, z)
#define R0F(x, z) ROF(x, 0, z)
#define trav(x, y) for(auto&x:y)
using ll = long long;
using vi = vector<int>;
using vl = vector<long long>;
using pii = pair<int, int>;
using vpii = vector<pair<int, int>>;
template<class T> inline bool ckmin(T&a, T b) {return b < a ? a = b, 1 : 0;}
template<class T> inline bool ckmax(T&a, T b) {return b > a ? a = b, 1 : 0;}
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
const char nl = '\n';
const int mxN = 1e5 + 10;
const int MOD = 1'000'000'009;
const long long infLL = 1e18;
int32_t valid(int n, int* _inputSeq){
vi inputSeq(n);
F0R(i, n){
inputSeq[i] = _inputSeq[i];
inputSeq[i]--;
}
vi vals;
F0R(i, n){
vals.pb(inputSeq[i]);
}
sort(all(vals));
vals.resize(unique(all(vals))-vals.begin());
if(siz(vals)!=n) return 0;
if(vals[0]>=n) return 1;
int pos;
F0R(i, n){
if(inputSeq[i] < n){
pos = i;
break;
}
}
int ff = inputSeq[pos];
F0R(i, n){
if(inputSeq[i]<n){
int dif = i - pos;
int should = (n + ff + dif)%n;
if(inputSeq[i] != should) return 0;
}
}
return 1;
}
int32_t replacement(int n, int _gondolaSeq[], int _replacementSeq[]){
vi gs(n), rs;
F0R(i, n){
gs[i] = _gondolaSeq[i]-1;
}
int pos = 0, cnt = 0, mx = -1, sp, mxpos;
sp = 0;
map<int, int> vals;
F0R(i, n){
if(ckmax(mx, gs[i])) mxpos = i;
if(gs[i] < n){
pos = i;
sp = gs[i];
cnt++;
}
}
if(cnt==n){
return 0;
}
int foo;
F0R(i, n){
int dif = i - pos;
int should= (n + sp + dif) % n;
if(mxpos==i){
foo = should;
}
if(gs[i]==should)
continue;
vals[gs[i]]=should;
}
int x = 0;
FOR(i, n, mx+1){
if(vals.find(i)!=vals.end()&&i!=mx){
_replacementSeq[x] = vals[i]+1;
x++;
continue;
}
_replacementSeq[x] = foo+1;
foo=i;
x++;
}
return x;
}
// modulo class
int mul(int x, int y){ return ((int64_t)x * (int64_t)y) % MOD; }
void ml(int &x, int y) {x = mul(x, y);}
int binpow(int x, int y){ int z = 1; while(y > 0) { if(y % 2 == 1) z = mul(z, x); x = mul(x, x); y /= 2; } return z; }
int32_t countReplacement(int n, int inputSeq[]){
if(!valid(n, inputSeq)) return 0;
int res = 1;
sort(inputSeq, inputSeq+n);
bool f = 1;
F0R(i, n){
if(inputSeq[i]>n){
if(f){
ml(res, binpow(n-i, inputSeq[i]-n-1));
}
else{
ml(res, binpow(n-i, inputSeq[i]-inputSeq[i-1]-1));
}
f = 0;
}
}
if(inputSeq[0]>n) ml(res, n);
return res;
}
