#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
pair<int, int> pii;
const int maxn = 1e5 + 100;
int n, d, c;
int used[maxn];
int f[maxn];
void move_inside(int i);
void move_outside(int i);
int press_button();
void add(int i){
used[i] = 1;
c += f[i];
move_inside(i);
}
void del(int i){
used[i] = 0;
c -= f[i];
move_outside(i);
}
void clear(){
for(int i = 1; i <= n; i++){
if(used[i]) del(i);
}
}
void calc(int x, int y){
if(x < y){
for(int i = 0; i < n; i++){
if(used[i]) continue;
add(i);
if(press_button() > y) del(i);
}
} else{
for(int i = n - 1; i >= 0; i--){
if(used[i] && press_button() > y){
del(i);
}
}
for(int i = 0; i < n; i++){
if(used[i]) continue;
add(i);
if(press_button() > y) del(i);
}
}
}
int min_cardinality(int N){
d = 0; n = N;
for(int i = N - 1; i >= 0; i--){
add(i);
if(press_button() == 1) f[i] = 1, d++;
else del(i);
} clear(); c = 0;
int ans = n, x = 0;
for(int l = 1, r = n - 1; l <= r;){
int mid = (l + r) >> 1;
calc(x, mid); x = mid;
if(!c) l = mid + 1;
else r = mid - 1, ans = mid;
}
return ans;
}
