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#include "gondola.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
typedef complex<ld> cd;
typedef pair<int, int> pi;
typedef pair<ll,ll> pl;
typedef pair<ld,ld> pd;
typedef vector<int> vi;
typedef vector<ld> vd;
typedef vector<ll> vl;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<cd> vcd;
template<class T> using pq = priority_queue<T>;
template<class T> using pqg = priority_queue<T, vector<T>, greater<T>>;
#define FOR(i, a, b) for (int i=a; i<(b); i++)
#define F0R(i, a) for (int i=0; i<(a); i++)
#define FORd(i,a,b) for (int i = (b)-1; i >= a; i--)
#define F0Rd(i,a) for (int i = (a)-1; i >= 0; i--)
#define trav(a,x) for (auto& a : x)
#define uid(a, b) uniform_int_distribution<int>(a, b)(rng)
#define sz(x) (int)(x).size()
#define mp make_pair
#define pb push_back
#define f first
#define s second
#define lb lower_bound
#define ub upper_bound
#define all(x) x.begin(), x.end()
#define ins insert
template<class T> bool ckmin(T& a, const T& b) { return b < a ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) { return a < b ? a = b, 1 : 0; }
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
const int MOD = 1000000009;
const char nl = '\n';
const int MX = 250001;
struct mi {
ll v; explicit operator ll() const { return v; }
mi() { v = 0; }
mi(ll _v) {
v = (-MOD < _v && _v < MOD) ? _v : _v % MOD;
if (v < 0) v += MOD;
}
friend bool operator==(const mi& a, const mi& b) {
return a.v == b.v; }
friend bool operator!=(const mi& a, const mi& b) {
return !(a == b); }
friend bool operator<(const mi& a, const mi& b) {
return a.v < b.v; }
mi& operator+=(const mi& m) {
if ((v += m.v) >= MOD) v -= MOD;
return *this; }
mi& operator-=(const mi& m) {
if ((v -= m.v) < 0) v += MOD;
return *this; }
mi& operator*=(const mi& m) {
v = v*m.v%MOD; return *this; }
mi& operator/=(const mi& m) { return (*this) *= inv(m); }
friend mi pow(mi a, ll p) {
mi ans = 1; assert(p >= 0);
for (; p; p /= 2, a *= a) if (p&1) ans *= a;
return ans;
}
friend mi inv(const mi& a) { assert(a.v != 0);
return pow(a,MOD-2); }
mi operator-() const { return mi(-v); }
mi& operator++() { return *this += 1; }
mi& operator--() { return *this -= 1; }
mi operator++(int) { mi temp; temp.v = v++; return temp; }
mi operator--(int) { mi temp; temp.v = v--; return temp; }
friend mi operator+(mi a, const mi& b) { return a += b; }
friend mi operator-(mi a, const mi& b) { return a -= b; }
friend mi operator*(mi a, const mi& b) { return a *= b; }
friend mi operator/(mi a, const mi& b) { return a /= b; }
friend ostream& operator<<(ostream& os, const mi& m) {
os << m.v; return os;
}
friend istream& operator>>(istream& is, mi& m) {
ll x; is >> x;
m.v = x;
return is;
}
};
typedef vector<mi> vmi;
typedef pair<mi,mi> pmi;
typedef vector<pmi> vpmi;
int valid(int N, int A[])
{
int val = -1;
set<int> found;
F0R(i, N) {
int X = A[i];
if (found.count(X)) return 0;
found.ins(X);
if (val == -1 && X <= N) {
val = (X - i + N) % N;
}
if (X <= N && val != (X - i + N) % N) return 0;
}
return 1;
}
//----------------------
int replacement(int N, int A[], int res[])
{
set<int> need;
map<int, int> rep;
int cur[N];
int p = 0;
F0R(i, N) {
if (A[i] <= N) p = (i - A[i] + N + 1) % N;
}
F0R(i, N) {
cur[i] = (i + N - p) % N + 1;
}
int hi = 0;
F0R(i, N) {
ckmax(hi, A[i]);
if (A[i] > N) {
need.ins(i); rep[A[i]] = i;
}
}
int ans = hi - N;
FOR(i, N+1, hi+1) {
if (rep.count(i)) {
res[i-N-1] = cur[rep[i]];
need.erase(rep[i]);
} else {
res[i-N-1] = cur[*need.begin()];
}
}
return ans;
}
//----------------------
int countReplacement(int N, int A[])
{
if (valid(N, A) == 0) return 0;
int cnt = 0;
int hi = 0;
F0R(i, N) {
if (A[i] > N) cnt++;
ckmax(hi, A[i]);
}
int num = hi - N;
mi ans = pow(mi(cnt), num-cnt);
if (cnt == N) {
ans *= N;
}
return (int) ans.v;
}
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