description: Let's call a set of positive integers a_1, a_2, ..., a_k quadratic if the product of the factorials of its elements is a square of an integer, i. e. ∏_{i=1}^{k} a_i! = m^2, for some integer m.
  
  You are given a positive integer n.
  
  Your task is to find a quadratic subset of a set 1, 2, ..., n of maximum size. If there are multiple answers, print any of them.
  
  Input
  
  A single line contains a single integer n (1 ≤ n ≤ 10^6).
  
  Output
  
  In the first line, print a single integer — the size of the maximum subset. In the second line, print the subset itself in an arbitrary order.
  
  Examples
  
  Input
  
  
  1
  
  
  Output
  
  
  1
  1 
  
  
  Input
  
  
  4
  
  
  Output
  
  
  3
  1 3 4 
  
  
  Input
  
  
  7
  
  
  Output
  
  
  4
  1 4 5 6 
  
  
  Input
  
  
  9
  
  
  Output
  
  
  7
  1 2 4 5 6 7 9 
solution: #include <bits/stdc++.h>
  using namespace std;
  const int N = 1000 * 1000 + 13;
  using li = unsigned long long;
  int pr[N];
  li hs[N], f[N];
  unordered_map<li, int> rf;
  int main() {
    int n;
    scanf("%d", &n);
    mt19937_64 rnd(chrono::steady_clock::now().time_since_epoch().count());
    iota(pr, pr + N, 0);
    for (int i = 2; i <= n; ++i)
      if (pr[i] == i) {
        for (int j = i; j <= n; j += i) pr[j] = min(pr[j], i);
        hs[i] = rnd();
      }
    for (int i = 2; i <= n; ++i) {
      f[i] = f[i - 1];
      int x = i;
      while (x != 1) {
        f[i] ^= hs[pr[x]];
        x /= pr[x];
      }
      rf[f[i]] = i;
    }
    auto solve = [&](int n) -> vector<int> {
      li fp = 0;
      for (int i = 2; i <= n; ++i) fp ^= f[i];
      if (fp == 0) return {};
      if (rf.count(fp)) return {rf[fp]};
      for (int i = 2; i <= n; ++i) {
        li x = f[i] ^ fp;
        if (rf.count(x) && rf[x] != x) return {rf[x], i};
      }
      return {2, n / 2, n};
    };
    auto ans = solve(n);
    printf("%d\n", n - (int)ans.size());
    for (int i = 1; i <= n; ++i)
      if (find(ans.begin(), ans.end(), i) == ans.end()) printf("%d ", i);
    puts("");
  }
  