#include "insects.h"
#include <bits/stdc++.h>

using namespace std;
using ll = long long;

mt19937 rnd(1337);

void razmeshay(vector<int> &a) {
    for (int t = 0; t < 3; ++t) {
        for (int i = 0; i < size(a); ++i) {
            swap(a[i], a[rnd() % size(a)]);
        }
    }
}

int min_cardinality(int N) {
    vector<int> a(N);
    iota(a.begin(), a.end(), 0);

    razmeshay(a);

    vector<int> leaders, inside(N);
    int maxSize = 0, T = 0, cntIn = 0, lastQueryT = 0;
    int queriesCnt[3]{};

    auto insert = [&](int x) -> bool {
        if (inside[x]) {
            return false;
        }
        cntIn += 1;
        move_inside(x);
        T += 1;
        queriesCnt[0] += 1;
        return inside[x] = true;
    };

    auto erase = [&](int x) -> void {
        if (!inside[x]) {
            return;
        }
        cntIn -= 1;
        move_outside(x);
        T += 1;
        queriesCnt[1] += 1;
        inside[x] = false;
    };

    auto query = [&]() -> int {
        if (T == lastQueryT) {
            return maxSize;
        }
        lastQueryT = T;
        queriesCnt[2] += 1;
        return maxSize = press_button();
    };

    vector<int> others;

    for (int x : a) {
        insert(x);
        if (query() == 1) {
            leaders.push_back(x);
        } else {
            others.push_back(x);
            erase(x);
        }
    };


    int ans = N;
    if (size(leaders) > size(others)) {
        return 1;
    } else if (size(leaders) == 1) {
        return N;
    }

    if (0) {
        shuffle(leaders.begin(), leaders.end(), rnd);
        auto dfs = [&](auto dfs, vector<int> lead, vector<int> oth, int isFull) -> void {
            if (ans == 1) {
                return;
            }
            if (size(lead) == 1) {
                ans = min<int>(ans, 1 + size(oth));
                return;
            }
            if (size(lead) > size(oth)) {
                ans = 1;
                return;
            }
            int mid = size(lead) / 2;
            vector<int> leadLeft(lead.begin(), lead.begin() + mid);
            vector<int> leadRight(lead.begin() + mid, lead.end());
            vector<int> nxt[2];
            if (isFull) {
                for (int x: leadRight) {
                    erase(x);
                }
            } else {
                for (int x: leadLeft) {
                    insert(x);
                }
            }
            for (int x: oth) {
                insert(x);
                if (query() == 1) {
                    nxt[1].push_back(x);
                } else {
                    nxt[0].push_back(x);
                }
                erase(x);
            }
            dfs(dfs, leadLeft, nxt[0], true);
            dfs(dfs, leadRight, nxt[1], false);
        };

        dfs(dfs, leaders, others, true);
        return ans;
    } else {
        shuffle(others.begin(), others.end(), rnd);
        razmeshay(others);
        int k = 4, i, cnt;
        for (k = 2, i = 0, cnt = 0; cnt < size(others) && k * size(leaders) <= N; i = i + 1 == size(others) ? 0 : i + 1) {
            if (insert(others[i])) {
                if (query() > k) {
                    erase(others[i]);
                } else {
                    if (size(leaders) * k == cntIn) {
                        k += 2;
                        cnt = 0;
                    }
                }
            }
            cnt += 1;
        }
         k = max(k - 3, 2);
            for (i = 0, cnt = 0; cnt < size(others) && k * size(leaders) <= N; i = i + 1 == size(others) ? 0 : i + 1) {
                if (insert(others[i])) {
                    if (query() > k) {
                        erase(others[i]);
                    } else {
                        if (size(leaders) * k == cntIn) {
                            k += 1;
                            cnt = 0;
                        }
                    }
                }
                cnt += 1;
            }
        return k - 1;
    }
}