#include "bits/extc++.h"
using namespace std;
template <typename T>
void dbgh(const T& t) {
cerr << t << endl;
}
template <typename T, typename... U>
void dbgh(const T& t, const U&... u) {
cerr << t << " | ";
dbgh(u...);
}
#ifdef DEBUG
#define dbg(...) \
cerr << "L" << __LINE__ << " [" << #__VA_ARGS__ << "]" \
<< ": "; \
dbgh(__VA_ARGS__)
#else
#define cerr \
if (false) \
cerr
#define dbg(...)
#endif
#define endl "\n"
#define long int64_t
#define sz(x) int(size(x))
void move_inside(int i);
void move_outside(int i);
int press_button();
int min_cardinality(int n) {
set<int> s;
auto move_in = [&](int i) -> void {
move_inside(i);
assert(!s.count(i));
s.insert(i);
};
auto move_out = [&](int i) -> void {
move_outside(i);
assert(s.count(i));
s.erase(i);
};
auto press_but = [&]() -> int {
assert(sz(s));
return press_button();
};
for (int i = 0; i < n; i++) {
move_in(i);
if (press_but() > 1) {
move_out(i);
}
}
int distinct_cnt = sz(s);
int l = 1, r = n / distinct_cnt;
while (l < r) {
int mid = max(l + 1, (3 * l + 1 * r) / 4);
vector<int> changed;
for (int i = 0; i < n; i++) {
if (!s.count(i)) {
move_in(i);
if (press_but() > mid) {
move_out(i);
} else {
changed.push_back(i);
}
}
}
if (sz(s) == distinct_cnt * mid) {
l = mid;
} else {
r = mid - 1;
for (auto& a : changed) {
move_out(a);
}
}
}
return l;
}
