#include "gondola.h"
#include <bits/stdc++.h>
#pragma GCC target("lzcnt,popcnt")
#pragma GCC optimize("O3,unroll-loops")
#define int long long
#define pii pair<int,int>
#define vi vector<int>
#define ff first
#define ss second
#define all(x) x.begin(),x.end()
#define sp << " " <<
using namespace std;
int32_t valid(int32_t n, int32_t inputSeq[])
{
map<int,int> saw;
vi pos(n+1,-1);
for (int i=0;i<n;i++) {
if (saw.count(inputSeq[i])) return 0;
saw[inputSeq[i]] = 1;
if (inputSeq[i] <= n) {
pos[inputSeq[i]] = i;
}
}
int frst = -1;
int lst = -1,lsty = -1;
for (int j = 1;j<=n;j++) {
if (pos[j] == -1) continue;
if (frst == -1) frst = pos[j];
pos[j] = (pos[j]-frst+n)%n;
if (lst != -1 && pos[j]-lst != j-lsty) return 0;
lst = pos[j];
lsty = j;
}
return 1;
}
//----------------------
int32_t replacement(int32_t n, int32_t gondolaSeq[], int32_t replacementSeq[])
{
int cur = n;
vi inds(n+1);
vi a(n+1);
iota(all(a),0ll);
int frst = -1;
for (int i=1;i<=n;i++) {
if (gondolaSeq[i-1] <= n) frst = i;
}
if (frst != -1) {
a[frst] = gondolaSeq[frst-1];
for (int j = frst+1;j<=n;j++) a[j] = a[j-1]%n+1;
for (int j = frst-1;j>=1;j--) a[j] = (a[j+1]==1)?n:(a[j+1]-1);
}
iota(all(inds),0ll);
sort(1+all(inds),[&](int x,int y) {
return gondolaSeq[x-1] < gondolaSeq[y-1];
});
int ptr = 0;
for (int i = 1;i<=n;i++) {
int ind = inds[i];
while (gondolaSeq[ind-1] > a[ind]) {
replacementSeq[ptr++] = a[ind];
a[ind] = ++cur;
}
}
return ptr;
}
//----------------------
const int MOD = 1e9+9;
int mult(int x,int y) {
return (x*y)%MOD;
}
int expo(int x,int y) {
if (!y) return 1;
int e = expo(x,y>>1);
e = mult(e,e);
if (y&1) e = mult(e,x);
return e;
}
int32_t countReplacement(int32_t n, int32_t inputSeq[])
{
int ans = valid(n,inputSeq);
vi inds(n+1);
iota(all(inds),0ll);
sort(1+all(inds),[&](int x,int y) {
return inputSeq[x-1] < inputSeq[y-1];
});
int rem = 0;
for (int i=1;i<=n;i++) if (inputSeq[i-1] > n) rem++;
int cur = n;
for (int i=1;i<=n;i++) {
if (inputSeq[ind-1] <= n) continue;
int ind = inds[i];
if (inputSeq[ind-1] <= cur) assert(0);
ans = mult(ans,expo(rem,(inputSeq[ind-1]-cur-1)));
cur = inputSeq[ind-1];
rem--;
}
return ans;
}
