Problem p02697 (C++) — buggy submission, verdict: Runtime Error

#include <algorithm>
#include <bits/stdc++.h>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <iostream>
#include <list>
#include <map>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdio.h>
#include <string>
#include <vector>
using namespace std;
typedef long long ll;
typedef long double ld;
using edge = struct {
  ll to;
  ll cost;
};
using point = struct {
  ll x;
  ll y;
};
typedef string str;
typedef std::pair<ll, ll> pl;
typedef std::pair<ll, pl> pl3;
typedef std::map<string, ll> msl;
typedef std::map<char, ll> mcl;
typedef std::map<ll, ll> mll;
typedef std::vector<ll> vl;
typedef std::vector<pl> vpl;
typedef std::vector<point> points;
typedef std::vector<pl3> vpl3;
typedef std::priority_queue<ll> pq;
typedef std::priority_queue<ll, vl, greater<ll>>
    pql; // priority queue taking from the lower value.
typedef std::vector<edge> gr;
const ll MOD = 1e9 + 7;
// const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-9;
const long double PI = 3.14159265358979323846;
vpl dirs = {
    {1, 0}, {0, 1},  {-1, 0},  {0, -1}, // four directions
    {1, 1}, {-1, 1}, {-1, -1}, {1, -1}, // diagonal
    {0, 0}                              // self
};

#define rep(i, n) for (ll(i) = 0; (i) < (n); (i)++)
#define revrep(i, n) for (ll(i) = n - 1; (i) >= 0; (i)--)
#define For(i, a, b) for (ll(i) = (a); (i) < (b); (i)++)
#define revFor(i, b, a) for (ll(i) = (b)-1; (i) >= (a); (i)--)
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define isUpper(c) ('a' - c > 0)
#define isNum(c) (0 <= (c) - '0' && (c) - '0' <= 9)
#define toLower(c) char((c) + 0x20)
#define toUpper(c) char((c)-0x20)
#define pb push_back
#define mp make_pair
#define pr(a) cout << (a)
#define prl(a) cout << (a) << endl
#define prl2(a, b) cout << (a) << " " << (b) << endl
#define prl3(a, b, c) cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a, b, c, d)                                                       \\
  cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) cout << (a) << " "
#define yn(condition)                                                          \\
  if ((condition))                                                             \\
    prl("Yes");                                                                \\
  else                                                                         \\
    prl("No");
#define YN(condition)                                                          \\
  if ((condition))                                                             \\
    prl("YES");                                                                \\
  else                                                                         \\
    prl("NO");
#define in1(a) cin >> (a)
#define in2(a, b) cin >> (a) >> (b)
#define in3(a, b, c) cin >> (a) >> (b) >> (c)
#define in4(a, b, c, d) cin >> (a) >> (b) >> (c) >> (d)
#define e1 first
#define e2 second
#define ctol(c) ll((c)) - ll('0')
#define ltos(n) to_string((n))
#define items(kv, v) for (auto &(kv) : (v))
#define ndig(N, n) ctol(ll(ltos((N))[ll(ltos((N)).length()) - (n)]))
#define rsort(a, n) sort(a, a + n, greater<>())
#define Forchar(c, a, z) for (char(c) = (a); (c) <= (z); (c)++)
#define cntchar(s, c) count(all((s)), c)
#define substring(s, start, end) s.substr((start), (end) - (start) + 1)
#define prl_nd(num, digits)                                                    \\
  cout << fixed << setprecision(digits) << (num) << endl;
#define XOR(a, b) (a) ^ (b)
#define prl_time(s)                                                            \\
  prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))
#define chmin(x, y) x = min(x, y)
#define chmax(x, y) x = max(x, y)
#define DEBUG(x) cerr << #x << ": " << (x) << endl
#define DEBUG_VEC(v)                                                           \\
  cerr << #v << ": ";                                                          \\
  rep(__i, (v).size()) cerr << ((v)[__i]) << ", ";                             \\
  cerr << endl

struct MaxFlow {
  struct F_edge {
    ll to, rev, capacity;
    F_edge(ll to, ll rev, ll capacity) : to(to), rev(rev), capacity(capacity) {}
  };

  typedef vector<F_edge> F_edges;
  vector<F_edges> graph;
  ll n_vertex;
  // level is the shortest path to get a given node from the source node.
  vl level, iter;

  MaxFlow(ll n_vertex) : n_vertex(n_vertex) { graph.resize(n_vertex); }

  void add_edge(ll from, ll to, ll capacity) {
    graph[from].pb({to, ll(graph[to].size()), capacity});
    graph[to].pb({from, ll(graph[from].size()) - 1, 0});
  }

  void bfs(ll source) {
    level = vl(n_vertex, -1);
    level[source] = 0;
    queue<ll> q;
    q.push(source);
    while (!q.empty()) {
      ll vertex = q.front();
      q.pop();
      rep(i, graph[vertex].size()) {
        ll target = graph[vertex][i].to;
        ll cap_target = graph[vertex][i].capacity;
        // if the flow can be into the target node, implement below.
        if (cap_target > 0 && level[target] < 0) {
          level[target] = level[vertex] + 1;
          q.push(target);
        }
      }
    }
  }

  ll dfs(ll vertex, ll sink, ll flow) {
    if (vertex == sink)
      return flow;
    for (ll &i = iter[vertex]; i < graph[vertex].size(); i++) {
      ll target = graph[vertex][i].to;
      ll cap_target = graph[vertex][i].capacity;
      ll rev_target = graph[vertex][i].rev;
      // if capasitiy is not full yet and target is farther,
      // then assign current flow
      if (cap_target > 0 && level[vertex] < level[target]) {
        ll d = dfs(target, sink, min(cap_target, flow));
        if (d > 0) { // if the flow successfully reaches the sink, reduce the
                     // flow from the capacity
          graph[vertex][i].capacity -= d;
          graph[target][rev_target].capacity += d;
          return d;
        }
      }
    }
    return 0;
  }

  ll dinic(ll source, ll sink) {
    ll flow = 0;
    while (true) {
      bfs(source);
      // if there is no path leading to the sink, the maximum flow is 0.
      if (level[sink] < 0)
        return flow;
      iter = vl(n_vertex, 0);
      ll f;
      while ((f = dfs(source, sink, INF)) > 0)
        flow += f;
    }
  }
};

class UnionFind {
  vl parents, set_size;

public:
  UnionFind() {}
  UnionFind(ll n) {
    parents = set_size = vl(n);
    rep(i, n) {
      parents[i] = i;
      set_size[i] = 1LL;
    }
  }

  ll root_find(ll x) {
    if (parents[x] == x)
      return x;
    return parents[x] = root_find(parents[x]);
  }

  void unite(ll x, ll y) {
    x = root_find(x);
    y = root_find(y);
    if (x == y)
      return;
    if (set_size[x] < set_size[y]) {
      parents[y] = x;
      set_size[x] += set_size[y];
    } else {
      parents[x] = y;
      set_size[y] += set_size[x];
    }
  }

  bool is_same(ll x, ll y) { // connected or not
    return root_find(x) == root_find(y);
  }

  ll size(ll x) { return set_size[root_find(x)]; }
};

/*
class LCA{
public:
    ll N, logN;
    vl depth, len;
    gr tree[200005]; // global declaration later.
    vector<vl> parents;

    LCA(ll n){
        N = n;
        logN = 0;
        while (N > (1LL << logN)) logN++;
        depth = vl(N); len = vl(N);
        parents = vector<vl>(logN, vl(N));
        init(0, -1, 0, 0);
        build();
    }

    void init(ll source, ll parent, ll d, ll l){
        depth[source] = d;
        parents[0][source] = parent;
        len[source] = l;
        rep(i, tree[source].size()){
            ll target = tree[source][i].to;
            ll cost = tree[source][i].cost;
            if (target == parent) continue;
            init(target, source, d + 1, cost + l);
        }
    }

    void build(){
        rep(k, logN - 1) rep(n, N){
            // if there is no parent, -1.
            // otherwise, the parent of the parent is the parent.
            if (parents[k][n] < 0) parents[k + 1][n] = -1;
            else parents[k + 1][n] = parents[k][parents[k][n]];
        }
    }

    ll query(ll u, ll v){
        if (depth[u] > depth[v]) swap(u, v);
        rep(k, logN) if ((depth[v] - depth[u]) >> k & 1) v = parents[k][v];
        if (u == v) return u;

        revrep(k, logN){
            if (parents[k][u] != parents[k][v]){
                u = parents[k][u]; v = parents[k][v];
            }
        }
        return parents[0][u];
    }

    ll distance(ll u, ll v){
        ll w = query(u, v);
        return len[u] + len[v] - 2 * len[w];
    }

};
*/

struct BIT {
  ll n;
  vl tree_dat;
  BIT(ll n) : tree_dat(n + 1, 0), n(n){};

  //       x: 1001 1010  1100 1011 1101  1111
  // x & - x: 0001 0010  0100 0001 0001  0001
  //      ->: 1010 1100 10000 1100 1100 10000

  ll update_func(ll val, ll dat) {
    // if maximum -> max(val, dat)
    // return max(val, dat);
    // if cumulative sum
    return val + dat;
  }

  ll query(ll i) {
    /*
    e.g.) i = 10101
    itr1. 10101 -> 10100
    itr2. 10100 -> 10000
    itr3. 10000 -> 00000 (break)
    */
    ll ret = 0;
    for (ll j = i; j >= 0; j = (j & (j + 1)) - 1) {
      ret = update_func(ret, tree_dat[j]);
    }
    return ret;
  }

  ll lower_bound(ll key) {
    if (key <= 0)
      return 0;
    ll left = 0, right = 1;
    while (right <= n)
      right *= 2;

    for (ll i = right; i > 0; i /= 2) {
      if (left + i <= n && tree_dat[left + i - 1] < key) {
        key -= tree_dat[left + i - 1];
        left += i;
      }
    }
    return left;
  }

  void update(ll i, ll val) {
    /*
    e.g.) i = 10101, n = 11111
    itr1. 10101 ->  10110
    itr2. 10110 ->  11000
    itr3. 11000 -> 100000 (break)
    */
    if (i < 0)
      return;
    for (ll j = i; j < n; j |= j + 1) {
      tree_dat[j] = update_func(val, tree_dat[j]);
    }
  }
};

ll gcd(ll m, ll n) {
  ll a = max(m, n);
  ll b = min(m, n);
  while (b != 1 && b != 0) {
    a %= b;
    swap(a, b);
  }
  return b == 1 ? 1 : a;
}

ll lcm(ll m, ll n) { return m / gcd(m, n) * n; }

ll power_mod(ll a, ll power, ll mod) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = (value * a) % mod;
    a = (a * a) % mod;
    power = power >> 1;
  }
  return value % mod;
}

ll modinv(ll a, ll mod) { return power_mod(a, mod - 2, mod); }

ll power_normal(ll a, ll power) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = value * a;
    a = a * a;
    power = power >> 1;
  }
  return value;
}

ll comb_memo[55][55];

ll pascal_triangle(ll n) {
  comb_memo[0][0] = 1;
  For(i, 1, n + 1) rep(j, i + 1) {
    comb_memo[i][j] += comb_memo[i - 1][j];
    if (j > 0)
      comb_memo[i][j] += comb_memo[i - 1][j - 1];
  }
}

ll combination(ll n, ll r, ll mod) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    ll num = (n - i) % mod, den = (i + 1) % mod;
    (numerator *= num) %= mod;
    (denomenator *= modinv(den, mod)) %= mod;
  }

  return (numerator * denomenator) % mod;
}

ll combination_memo(ll n, ll r, ll pre, ll mod) {
  // pre = nCr-1
  // return nCr
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = ll(n - r + 1) % mod;
  ll denomenator = modinv(r, mod) % mod;
  ll val = (numerator * denomenator) % mod;
  val *= pre;
  return val % mod;
}

ll combination_no_mod(ll n, ll r) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    numerator *= n - i;
    denomenator *= i + 1;
    ll g = gcd(numerator, denomenator);
    numerator /= g;
    denomenator /= g;
  }

  return numerator;
}

ld log_combination(ll n, ll r) {
  if (n == r && n == 0)
    return 0;
  else if (n <= 0 || r < 0 || r > n)
    return -INF;
  ld val = 0;
  for (ll i = 0; i < r; i++) {
    val += log(n - i);
    val -= log(i + 1);
  }

  return val;
}

ll bin_search(ll key, ll A[], ll left, ll right) {
  // return the index idx where A[idx] = key.
  // A[left] is start and A[right] is end..
  // In other words, A[right], not A[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (A[mid] == key)
      return mid;
    else if (A[mid] > key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid + 1;
  }
  return -1;
}

/*
ll bin_search_temp(ll left, ll right, callable judge){
    while(right > left){
        // when seeking lower bound
        ll mid = (right + left) / 2;
        if (judge(mid)) right = mid;
        else left = mid + 1;
        // when seeking upper bound
        ll mid = (right + left + 1) / 2;
        if (judge(mid)) left = mid;
        else right = mid - 1;
    }

    return right;
}

trinary_search

// Care the value EPS!!! Compare to the condition
ld left = 0; ld right = p;
while(abs(right - left) > EPS){
    ld left2 = (2 * left + right) / 3.0;
    ld right2 = (left + 2 * right) / 3.0;
    ld f1 = tak_func(left2);
    ld f2 = tak_func(right2);
    if (f1 <= f2) right = right2;
    else if (f2 <= f1) left = left2;
}

*/

ll lower_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] >= key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] < key && key <= A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] < key && key <= A[mid])
      return mid;
    else if (A[mid - 1] >= key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid;
  }

  return -1; // all the elements < key
}

ll inf_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] > key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] <= key && key < A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] <= key && key < A[mid])
      return mid;
    else if (A[mid - 1] > key)
      right = mid - 1;
    else if (A[mid] <= key)
      left = mid;
  }

  return -1; // all the elements <= key
}

ll upper_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] <= key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] <= key && key < A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] <= key && key < A[mid + 1])
      return mid;
    else if (A[mid] > key)
      right = mid;
    else if (A[mid + 1] <= key)
      left = mid + 1;
  }

  return -1; // all the elements > key
}

ll sup_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] < key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] < key && key <= A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] < key && key <= A[mid + 1])
      return mid;
    else if (A[mid] >= key)
      right = mid;
    else if (A[mid + 1] < key)
      left = mid + 1;
  }

  return -1; // all the elements >= key
}

ll bin_search_vector(ll key, vl v, ll left, ll right) {
  // return the index idx where v[idx] = key.
  // v[left] is start and v[right] is end..
  // In other words, v[right], not v[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (v[mid] == key)
      return mid;
    else if (v[mid] > key)
      right = mid - 1;
    else if (v[mid] < key)
      left = mid + 1;
  }
  return -1;
}

ll lower_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] < key <= v[N]
  // N == -1 if all the elements < key
  return lower_bound(all(v), key) - v.begin();
}

ll inf_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] <= key < v[N] <= key + 1
  // N == -1 if all the elements <= key
  return lower_bound(all(v), key + 1) - v.begin();
}

ll upper_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key < v[N + 1]
  // N == -1 if all the elements > key
  return upper_bound(all(v), key) - v.begin(); // (- 1)
}

ll sup_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key - 1 < key <= v[N + 1]
  // N == -1 if all the elements >= key
  return upper_bound(all(v), key - 1) - v.begin() - 1;
}

ll fact(ll n) {
  if (n == 0)
    return 1;
  return n * fact(n - 1);
}

ll fact_mod(ll n, ll mod) {
  if (n == 0)
    return 1;
  return n * fact_mod(n - 1, mod);
}

bool is_prime(ll n) {
  if (n <= 1)
    return false;
  for (ll i = 2; i * i <= n; i++) {
    if (n % i == 0)
      return false;
  }
  return true;
}

ll bool_sum(ll a1, ll a2) {
  if (a1 == 1 || a2 == 1)
    return 1;
  return 0;
}

mll prime_factorization(ll n) {
  ll i = 2;
  mll table;
  while (i * i <= n) {
    while (n % i == 0) {
      table[i]++;
      n /= i;
    }
    i++;
  }
  if (n > 1)
    table[n] = 1;
  return table;
}

vl divisor_table(ll n) {
  vl table;
  ll i = 1;
  while (i * i <= n) {
    if (n % i == 0) {
      table.pb(i);
      if (i * i != n)
        table.pb(n / i);
    }
    i++;
  }

  sort(all(table));

  return table;
}

ll char_to_idx(char c) {
  ll idx = 0;
  Forchar(cc, 'a', 'z') {
    if (c == cc)
      return idx;
    else
      idx++;
  }
}

ll next_combination(ll sub) {
  /*
  ll n, k; ll bit = (1 << k) - 1;
  for (; bit < (1 << n); bit = next_combination(bit)){
      bool ith = bit & (1 << i);
      procedures...
  }
  */
  ll x = sub & -sub, y = sub + x;
  return (((sub & ~y) / x) >> 1) | y;
}

vl z_algorithm(str s) {
  ll n = s.length();
  vl res(n);
  res[0] = n;
  ll i1 = 1, i2 = 0;

  while (i1 < n) {
    while (i1 + i2 < n && s[i2] == s[i1 + i2])
      ++i2;
    res[i1] = i2;
    if (i2 == 0) {
      ++i1;
      continue;
    }
    ll i3 = 1;
    while (i1 + i3 < n && i3 + res[i3] < i2) {
      res[i1 + i3] = res[i3];
      ++i3;
    }
    i1 += i3, i2 -= i3;
  }
  return res;
}

ll big_number_mod(str s, ll mod) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth) % mod;
    val %= mod;
    tenth *= 10;
    tenth %= mod;
    idx++;
  }

  return val;
}

str bin_expression(ll n, ll dig) {
  str s = "";
  rep(i, dig) {
    s += ltos(n % 2);
    n /= 2;
  }
  reverse(all(s));
  return s;
}

ll string_to_ll(str s) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth);
    tenth *= 10;
    idx++;
  }

  return val;
}

str reflected_string(str s) {
  str t, u;
  ll n = s.length();
  t = s;
  reverse(all(t));
  u = substring(t, 0, n - 2) + s + substring(t, 1, n - 1);
  return u;
}

ld distance_between_point_line(point l_begin, point l_end, point p) {
  ll xl1 = l_begin.x, yl1 = l_begin.y;
  ll xl2 = l_end.x, yl2 = l_end.y;
  ll xp = p.x, yp = p.y;
  ll a = yl2 - yl1;
  ll b = -xl2 + xl1;
  ll c = -a * xl2 - b * yl2;
  return abs(ld(a * xp + b * yp + c)) / ld(sqrt(a * a + b * b));
}

bool is_cross(point l1_begin, point l1_end, point l2_begin, point l2_end) {
  ll x1 = l1_begin.x, y1 = l1_begin.y;
  ll x2 = l1_end.x, y2 = l1_end.y;
  ll x3 = l2_begin.x, y3 = l2_begin.y;
  ll x4 = l2_end.x, y4 = l2_end.y;
  ll val1 = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3);
  ll val2 = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4);
  ll val3 = (x3 - x4) * (y1 - y3) + (y3 - y4) * (x3 - x1);
  ll val4 = (x3 - x4) * (y2 - y3) + (y3 - y4) * (x3 - x2);

  return val1 * val2 < 0 && val3 * val4 < 0;
}

ld space_of_triangle(point p1, point p2, point p3) {
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;
  ll v1 = x2 - x1;
  ll u1 = y2 - y1;
  ll v2 = x3 - x1;
  ll u2 = y3 - y1;
  ld s = ld(v1 * u2 - u1 * v2) / ld(2);
  return abs(s);
}

pair<point, ll> center_2det_of_3points(point p1, point p2, point p3) {
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;
  ll c2x_2 = x2 + x1, c2y_2 = y2 + y1;
  ll c3x_2 = x3 + x1, c3y_2 = y3 + y1;
  ll b2y = x2 - x1, b2x = -y2 + y1;
  ll b3y = x3 - x1, b3x = -y3 + y1;
  ll x1_2 = c3x_2 * b3y - c3y_2 * b3x, y1_2 = c2x_2 * b2y - c2y_2 * b2x;
  ll det = -b3y * b2x + b3x * b2y;
  if (det == 0)
    return {{INF, INF}, det};
  ll cx_2det = -b2x * x1_2 + b3x * y1_2, cy_2det = -b2y * x1_2 + b3y * y1_2;
  return {{cx_2det, cy_2det}, det};
}

ll inversion_number(vl a, ll a_max) {
  /*
  Paramters
  ---------
  a: vector<ll>
      All the elements must be non-negative.
      Prefably the elements are compressed to reduce the computational cost.
  a_max: ll
      The maximum value of the vector a or the value bigger than the value
  stated previously.
  */

  BIT bit(a_max + 1);
  ll val = 0;
  rep(i, a.size()) {
    // i is the number of elements that have lower index than a[i].
    // call the number of elements that have lower value than a[i]
    // by subtracting these two, the residual number is the number of elements
    // that have larger value
    val += i - bit.query(a[i] - 1); // cumulative sum from 0 to a[i] - 1
    bit.update(a[i], 1);
  }
  return val;
}

template <typename T> vector<T> compress(vector<T> v) {
  // sort and remove all the duplicated values
  sort(all(v));
  v.erase(unique(all(v)), v.end());
  return v;
}

template <typename T> map<T, ll> dict(const vector<T> &v) {
  map<T, ll> d;
  rep(i, v.size()) d[v[i]] = i;
  return d;
}

/*
const ll N_VERTEX = 310;
ll a, b, t;
ll dist[N_VERTEX][N_VERTEX];

void warshall_floyd(ll n){
    // rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(k, n) rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][k] +
dist[k][j]);
}

int main(void){
    in2(n, m);
    rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        dist[a][b] = t;
        dist[b][a] = t;
    }

    warshall_floyd(n);
}

ll dist[1005], vertex_pre[1005];

void dijkstra(ll start, ll n) {
    priority_queue<pl, vector<pl>, greater<pl>> edge_costs;
    fill(dist, dist + n, INF); // fill(vertex_pre, vertex_pre + n, -1);
    dist[start] = 0;
    edge_costs.push(pl(0, start));

    while (!edge_costs.empty()) {
        pl edge_cost = edge_costs.top();
        edge_costs.pop();
        ll idx = edge_cost.second;
        ll cost = edge_cost.first;
        if (dist[idx] < cost) continue;

        rep(i, graph[idx].size()){
            edge e = graph[idx][i];
            if (dist[e.to] > dist[idx] + e.cost){
                dist[e.to] = dist[idx] + e.cost;
                // vertex_pre[e.to] = idx;
                edge_costs.push(pl(dist[e.to], e.to));
            }
        }
    }
}

vl get_predecessor(ll g){
    vl path;
    for (; g != -1; g = vertex_pre[g]) path.pb(g);

    reverse(all(path));
    return path;
}

int main(void){
    in2(n, m);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        G[a].pb({b, t});
        G[b].pb({a, t});
    }

    dijkstra(0, n);
}


# ABC061D

bool find_negative_cycle(ll goal){
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[e.to] != INF && dist[e.to] > dist[v] + e.cost){
            dist[e.to] = -INF;
            if (goal == -1) return true;
            else if (goal == e.to) return true;
        }
    }
    return false;
}

bool bellman_ford(ll start, ll n, ll goal){
    // if there is a closed circuit, it returns false. (when goal == -1)
    // if the distance to goal cannot be obtained, it returns false (when goal
!= -1) fill(dist, dist + n, INF); dist[start] = 0; rep(i, n) rep(v, n) rep(k,
graph[v].size()){ edge e = graph[v][k]; if (dist[v] != INF && dist[e.to] >
dist[v] + e.cost) dist[e.to] = dist[v] + e.cost;
    }

    if (find_negative_cycle(goal)) return false;
    return true;
}

*/

/*
    # 1. The usage of map pair
    map<pl, ll> cnt;
    cnt[{i, j}] = 0;
    items(kv, cnt){
        prl2(kv.first, kv.second);
    }

    # 2. The usage of next_permutation and combination (factorial search)
    ll a[8];
    rep(i, 8) a[i] = i;
    sort(a, a + 8);
    do{
    }while(next_permutation(a, a+n));

    // here, combination
    ll n, r;
    ll i1, i2, ..., ir;
    For(i1, n - r, n) For(i2, n - r - 1, i1) ... For(ir, n - 2 * r + 1, irr){
   process;}

    # 3. bit search
    ll n;
    in1(n);
    const ll base = 3;
    ll upper = power_normal(base, n);

    rep(i, upper){
        ll tmp = i;
        rep(j, n){
            rep(k, base) if (tmp % base == k) prl(k);
            tmp /= base;
        }
    }

    # 4. imos method
    // used when we would like to count the number which
    // shows how many times the numbers between l and r belongs to smt.
    // This method is composed of three process.
    ll n, m, s[MAX_M], l, r;
    in2(n, m);
    rep(i, m) s[i] = 0;

    // 1st step
    rep(i, n){
        in3(l, r, c);
        l--; r--; // if l starts from 1.
        s[l] += c; s[r + 1] -= c;
    }

    // 2nd step
    rep(i, m - 1) s[i + 1] += s[i];
    // 3rd step: judgement...

    #5. shakutori method (syakutori)
    // 1. strech right side while the condition is met.
    // 2. renew the answer
    // 3. increments left side
    // 4. Back to 1. (l <= r must be satisfied all the time.)

    ll l = 0; ll r = 0;
    while (l < n){
        r = max(r, l);
        if (l == r) r++;
        while(r < n && cond) r++;
        answer += r - l; l++;
    }
    prl(answer);

    #6. priority queue

    pq q;
    ll answer = 0;
    ll v;

    rep(i, n) q.push(a[i]);
    rep(i, m){
        v = q.top(); q.pop(); // get the top value and dump the value from queue
        v /= 2; q.push(v); // add the new value
    }

    while(!q.empty()){
        answer += q.top();
        q.pop();
    }

    #7. The shortest path (distance) between the k-th edge and another edge
   (Tree) ll depth[MAX_N]; gr tree[MAX_N];

    void dfs(ll source, ll parent, ll all_cost){
        depth[source] = all_cost;
        items(e, tree[source]){
            if (e.to == parent) continue;
            dfs(e.to, source, all_cost + e.cost);
        }
    }

    ll n, k, a, b, c;

    in2(n, k);
    rep(i, n - 1){
        in3(a, b, c);
        a--; b--;
        tree[a].pb({b, c});
        tree[b].pb({a, c});
    }

    k--;
    dfs(k, -1, 0);

    #10. Visiting Subtree using recurrent function (ABC138D)

    gr tree[MAX_N];
    ll c[MAX_N];
    bool visited[MAX_N];

    void dfs(ll source, ll parent, ll val){
        visited[source] = true;
        c[source] += val;
        rep(i, tree[source].size()){
            rep(i, m){ll res = n % match[i].e1;}
            ll vertex = tree[source][i].to;
            if (vertex == parent) continue;
            dfs(vertex, source, c[source]);
        }
    }

    #11. bfs ABC146D, ABC007C
    1. first create a tree.
    2. start searching from a node.
    3. do some processes and push nodes connected with a given target node in
   BFS.
    4. repeat a series of procedure until queue is empty.

    queue<pl> q;

    void bfs(ll source, ll parents){
        ll n_edge = G[source].size();
        if (parents != -1) dist[source] = min(dist[source], dist[parents] + 1);
        if (visited[source]) return;
        visited[source] = true;

        rep(idx, n_edge){
            ll target = G[source][idx].to;
            if (target == parents) continue;
            q.push(mp(target, source));
        }
    }

    q.push(mp(sg.e1, -1));
    while(!q.empty()){
        pl source = q.front(); q.pop();
        bfs(source.e1, source.e2);
    }

    #12. grid to distance matrix (dx, dy)
    ll w, h;
    ll pos_to_idx(ll x, ll y){
        return y * w + x;
    }

    pl idx_to_pos(ll idx){
        return mp(idx % w, idx / w);
    }

    int main(void){
        in2(h, w);
        rep(y, h){
            in1(s);
            rep(x, w){
                if (s[x] == '#') wall[x][y] = true;
                else wall[x][y] = false;
            }
        }

        rep(i1, h * w)rep(i2, h * w) dist[i1][i2] = INF * (i1 != i2);

        rep(x, w)rep(y, h){
            ll idx1 = pos_to_idx(x, y); ll idx2;
            if (wall[x][y]) continue;
            if (x != 0 && !wall[x - 1][y]){
                idx2 = pos_to_idx(x - 1, y);
                // if warshall floyd
                dist[idx1][idx2] = 1;
                // if dijkstra
                // graph[idx1].pb({idx2, 1});
            }
            if (x != w - 1 && !wall[x + 1][y]){
                idx2 = pos_to_idx(x + 1, y);
                dist[idx1][idx2] = 1;
                // graph[idx1].pb({idx2, 1});
            }
            if (y != 0 && !wall[x][y - 1]){
                idx2 = pos_to_idx(x, y - 1);
                dist[idx1][idx2] = 1;
                // graph[idx1].pb({idx2, 1});
            }
            if (y != h - 1 && !wall[x][y + 1]){
                idx2 = pos_to_idx(x, y + 1);
                dist[idx1][idx2] = 1;
                // graph[idx1].pb({idx2, 1});
            }
        }
    }


*/

/*
# the operators regarding bit
& (AND), | (OR), ^ (XOR)
- (REVERSE), >> (SMALLER SHIFT)
<< (BIGGER SHIFT)


     x1: 0000 0001 0010 0101 0110 0111 0111
     x2: xxxx 0001 0011 0100 0101 1000 0110
x1 & x2: 0000 0001 0010 0100 0100 0000 0110

      x: 1001 1010  1100 1011 1101  1111
x & - x: 0001 0010  0100 0001 0001  0001
    sum: 1010 1100 10000 1100 1100 10000

x << y is x * 2 ** y
x >> y is rep(i, y) x = x // 2

####### Attention #######
1 << i and 1LL << i is different. If programs show WA, try switch to this one.
Let S be a bit sequence and i be a non-negative integer
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & -(1 << i) -> S - {i}
__builtin_popcount(S) -> the number of elements in S
S = 0 -> S is an empty set
__builtin_popcountl(i) -> the number of 1 in binary
S = (1 << n) - 1 -> S includes all the elements up to the n-th

#Conditional Operator
condition ? true : false;

#iterator
type declaration: auto
value reference: *itr
increment: itr++
decrement: itr--
substitution of value: *itr = smt

# inclusion-exclusion principle
|U[i = 1 to n] Ai| = sum[i = 1 to n] |Ai| - sum[i < j]|Ai ^ Aj| + ... + (-1)^(n
- 1) |^[i = 1 to n]Ai|
*/

const ll MAX_N = 200005;
bool okay = false;
ll answer = 0;
ll n, m, a, b, c, d;

void solve() {
  if (n % 2) {
    rep(i, m) prl2(i + 1, n - i);
    return;
  }

  b = n / 2 - 1, c = 1;
  rep(i, (b + 1) / 2) {
    ll j = i + 1;
    if (c > m)
      break;
    prl2(j, b - j + 2);
    c++;
  }
  rep(i, (b + 1) / 2) {
    ll j = i + 1;
    if (c > m)
      break;
    prl2(j + b + 1, 2 * b - j + 2);
    c++;
  }
  // check negative MOD
  // check index flow
  // check overwrite of the input variables
}

int main(void) {
  in2(n, m);
  // assert(n <= 400);
  solve();

  return 0;
}


Error: -8

Fixed submission:

#include <algorithm>
#include <bits/stdc++.h>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <iostream>
#include <list>
#include <map>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdio.h>
#include <string>
#include <vector>
using namespace std;
typedef long long ll;
typedef long double ld;
using edge = struct {
  ll to;
  ll cost;
};
using point = struct {
  ll x;
  ll y;
};
typedef string str;
typedef std::pair<ll, ll> pl;
typedef std::pair<ll, pl> pl3;
typedef std::map<string, ll> msl;
typedef std::map<char, ll> mcl;
typedef std::map<ll, ll> mll;
typedef std::vector<ll> vl;
typedef std::vector<pl> vpl;
typedef std::vector<point> points;
typedef std::vector<pl3> vpl3;
typedef std::priority_queue<ll> pq;
typedef std::priority_queue<ll, vl, greater<ll>>
    pql; // priority queue taking from the lower value.
typedef std::vector<edge> gr;
const ll MOD = 1e9 + 7;
// const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-9;
const long double PI = 3.14159265358979323846;
vpl dirs = {
    {1, 0}, {0, 1},  {-1, 0},  {0, -1}, // four directions
    {1, 1}, {-1, 1}, {-1, -1}, {1, -1}, // diagonal
    {0, 0}                              // self
};

#define rep(i, n) for (ll(i) = 0; (i) < (n); (i)++)
#define revrep(i, n) for (ll(i) = n - 1; (i) >= 0; (i)--)
#define For(i, a, b) for (ll(i) = (a); (i) < (b); (i)++)
#define revFor(i, b, a) for (ll(i) = (b)-1; (i) >= (a); (i)--)
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define isUpper(c) ('a' - c > 0)
#define isNum(c) (0 <= (c) - '0' && (c) - '0' <= 9)
#define toLower(c) char((c) + 0x20)
#define toUpper(c) char((c)-0x20)
#define pb push_back
#define mp make_pair
#define pr(a) cout << (a)
#define prl(a) cout << (a) << endl
#define prl2(a, b) cout << (a) << " " << (b) << endl
#define prl3(a, b, c) cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a, b, c, d)                                                       \\
  cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) cout << (a) << " "
#define yn(condition)                                                          \\
  if ((condition))                                                             \\
    prl("Yes");                                                                \\
  else                                                                         \\
    prl("No");
#define YN(condition)                                                          \\
  if ((condition))                                                             \\
    prl("YES");                                                                \\
  else                                                                         \\
    prl("NO");
#define in1(a) cin >> (a)
#define in2(a, b) cin >> (a) >> (b)
#define in3(a, b, c) cin >> (a) >> (b) >> (c)
#define in4(a, b, c, d) cin >> (a) >> (b) >> (c) >> (d)
#define e1 first
#define e2 second
#define ctol(c) ll((c)) - ll('0')
#define ltos(n) to_string((n))
#define items(kv, v) for (auto &(kv) : (v))
#define ndig(N, n) ctol(ll(ltos((N))[ll(ltos((N)).length()) - (n)]))
#define rsort(a, n) sort(a, a + n, greater<>())
#define Forchar(c, a, z) for (char(c) = (a); (c) <= (z); (c)++)
#define cntchar(s, c) count(all((s)), c)
#define substring(s, start, end) s.substr((start), (end) - (start) + 1)
#define prl_nd(num, digits)                                                    \\
  cout << fixed << setprecision(digits) << (num) << endl;
#define XOR(a, b) (a) ^ (b)
#define prl_time(s)                                                            \\
  prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))
#define chmin(x, y) x = min(x, y)
#define chmax(x, y) x = max(x, y)
#define DEBUG(x) cerr << #x << ": " << (x) << endl
#define DEBUG_VEC(v)                                                           \\
  cerr << #v << ": ";                                                          \\
  rep(__i, (v).size()) cerr << ((v)[__i]) << ", ";                             \\
  cerr << endl

struct MaxFlow {
  struct F_edge {
    ll to, rev, capacity;
    F_edge(ll to, ll rev, ll capacity) : to(to), rev(rev), capacity(capacity) {}
  };

  typedef vector<F_edge> F_edges;
  vector<F_edges> graph;
  ll n_vertex;
  // level is the shortest path to get a given node from the source node.
  vl level, iter;

  MaxFlow(ll n_vertex) : n_vertex(n_vertex) { graph.resize(n_vertex); }

  void add_edge(ll from, ll to, ll capacity) {
    graph[from].pb({to, ll(graph[to].size()), capacity});
    graph[to].pb({from, ll(graph[from].size()) - 1, 0});
  }

  void bfs(ll source) {
    level = vl(n_vertex, -1);
    level[source] = 0;
    queue<ll> q;
    q.push(source);
    while (!q.empty()) {
      ll vertex = q.front();
      q.pop();
      rep(i, graph[vertex].size()) {
        ll target = graph[vertex][i].to;
        ll cap_target = graph[vertex][i].capacity;
        // if the flow can be into the target node, implement below.
        if (cap_target > 0 && level[target] < 0) {
          level[target] = level[vertex] + 1;
          q.push(target);
        }
      }
    }
  }

  ll dfs(ll vertex, ll sink, ll flow) {
    if (vertex == sink)
      return flow;
    for (ll &i = iter[vertex]; i < graph[vertex].size(); i++) {
      ll target = graph[vertex][i].to;
      ll cap_target = graph[vertex][i].capacity;
      ll rev_target = graph[vertex][i].rev;
      // if capasitiy is not full yet and target is farther,
      // then assign current flow
      if (cap_target > 0 && level[vertex] < level[target]) {
        ll d = dfs(target, sink, min(cap_target, flow));
        if (d > 0) { // if the flow successfully reaches the sink, reduce the
                     // flow from the capacity
          graph[vertex][i].capacity -= d;
          graph[target][rev_target].capacity += d;
          return d;
        }
      }
    }
    return 0;
  }

  ll dinic(ll source, ll sink) {
    ll flow = 0;
    while (true) {
      bfs(source);
      // if there is no path leading to the sink, the maximum flow is 0.
      if (level[sink] < 0)
        return flow;
      iter = vl(n_vertex, 0);
      ll f;
      while ((f = dfs(source, sink, INF)) > 0)
        flow += f;
    }
  }
};

class UnionFind {
  vl parents, set_size;

public:
  UnionFind() {}
  UnionFind(ll n) {
    parents = set_size = vl(n);
    rep(i, n) {
      parents[i] = i;
      set_size[i] = 1LL;
    }
  }

  ll root_find(ll x) {
    if (parents[x] == x)
      return x;
    return parents[x] = root_find(parents[x]);
  }

  void unite(ll x, ll y) {
    x = root_find(x);
    y = root_find(y);
    if (x == y)
      return;
    if (set_size[x] < set_size[y]) {
      parents[y] = x;
      set_size[x] += set_size[y];
    } else {
      parents[x] = y;
      set_size[y] += set_size[x];
    }
  }

  bool is_same(ll x, ll y) { // connected or not
    return root_find(x) == root_find(y);
  }

  ll size(ll x) { return set_size[root_find(x)]; }
};

/*
class LCA{
public:
    ll N, logN;
    vl depth, len;
    gr tree[200005]; // global declaration later.
    vector<vl> parents;

    LCA(ll n){
        N = n;
        logN = 0;
        while (N > (1LL << logN)) logN++;
        depth = vl(N); len = vl(N);
        parents = vector<vl>(logN, vl(N));
        init(0, -1, 0, 0);
        build();
    }

    void init(ll source, ll parent, ll d, ll l){
        depth[source] = d;
        parents[0][source] = parent;
        len[source] = l;
        rep(i, tree[source].size()){
            ll target = tree[source][i].to;
            ll cost = tree[source][i].cost;
            if (target == parent) continue;
            init(target, source, d + 1, cost + l);
        }
    }

    void build(){
        rep(k, logN - 1) rep(n, N){
            // if there is no parent, -1.
            // otherwise, the parent of the parent is the parent.
            if (parents[k][n] < 0) parents[k + 1][n] = -1;
            else parents[k + 1][n] = parents[k][parents[k][n]];
        }
    }

    ll query(ll u, ll v){
        if (depth[u] > depth[v]) swap(u, v);
        rep(k, logN) if ((depth[v] - depth[u]) >> k & 1) v = parents[k][v];
        if (u == v) return u;

        revrep(k, logN){
            if (parents[k][u] != parents[k][v]){
                u = parents[k][u]; v = parents[k][v];
            }
        }
        return parents[0][u];
    }

    ll distance(ll u, ll v){
        ll w = query(u, v);
        return len[u] + len[v] - 2 * len[w];
    }

};
*/

struct BIT {
  ll n;
  vl tree_dat;
  BIT(ll n) : tree_dat(n + 1, 0), n(n){};

  //       x: 1001 1010  1100 1011 1101  1111
  // x & - x: 0001 0010  0100 0001 0001  0001
  //      ->: 1010 1100 10000 1100 1100 10000

  ll update_func(ll val, ll dat) {
    // if maximum -> max(val, dat)
    // return max(val, dat);
    // if cumulative sum
    return val + dat;
  }

  ll query(ll i) {
    /*
    e.g.) i = 10101
    itr1. 10101 -> 10100
    itr2. 10100 -> 10000
    itr3. 10000 -> 00000 (break)
    */
    ll ret = 0;
    for (ll j = i; j >= 0; j = (j & (j + 1)) - 1) {
      ret = update_func(ret, tree_dat[j]);
    }
    return ret;
  }

  ll lower_bound(ll key) {
    if (key <= 0)
      return 0;
    ll left = 0, right = 1;
    while (right <= n)
      right *= 2;

    for (ll i = right; i > 0; i /= 2) {
      if (left + i <= n && tree_dat[left + i - 1] < key) {
        key -= tree_dat[left + i - 1];
        left += i;
      }
    }
    return left;
  }

  void update(ll i, ll val) {
    /*
    e.g.) i = 10101, n = 11111
    itr1. 10101 ->  10110
    itr2. 10110 ->  11000
    itr3. 11000 -> 100000 (break)
    */
    if (i < 0)
      return;
    for (ll j = i; j < n; j |= j + 1) {
      tree_dat[j] = update_func(val, tree_dat[j]);
    }
  }
};

ll gcd(ll m, ll n) {
  ll a = max(m, n);
  ll b = min(m, n);
  while (b != 1 && b != 0) {
    a %= b;
    swap(a, b);
  }
  return b == 1 ? 1 : a;
}

ll lcm(ll m, ll n) { return m / gcd(m, n) * n; }

ll power_mod(ll a, ll power, ll mod) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = (value * a) % mod;
    a = (a * a) % mod;
    power = power >> 1;
  }
  return value % mod;
}

ll modinv(ll a, ll mod) { return power_mod(a, mod - 2, mod); }

ll power_normal(ll a, ll power) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = value * a;
    a = a * a;
    power = power >> 1;
  }
  return value;
}

ll comb_memo[55][55];

ll pascal_triangle(ll n) {
  comb_memo[0][0] = 1;
  For(i, 1, n + 1) rep(j, i + 1) {
    comb_memo[i][j] += comb_memo[i - 1][j];
    if (j > 0)
      comb_memo[i][j] += comb_memo[i - 1][j - 1];
  }
}

ll combination(ll n, ll r, ll mod) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    ll num = (n - i) % mod, den = (i + 1) % mod;
    (numerator *= num) %= mod;
    (denomenator *= modinv(den, mod)) %= mod;
  }

  return (numerator * denomenator) % mod;
}

ll combination_memo(ll n, ll r, ll pre, ll mod) {
  // pre = nCr-1
  // return nCr
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = ll(n - r + 1) % mod;
  ll denomenator = modinv(r, mod) % mod;
  ll val = (numerator * denomenator) % mod;
  val *= pre;
  return val % mod;
}

ll combination_no_mod(ll n, ll r) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    numerator *= n - i;
    denomenator *= i + 1;
    ll g = gcd(numerator, denomenator);
    numerator /= g;
    denomenator /= g;
  }

  return numerator;
}

ld log_combination(ll n, ll r) {
  if (n == r && n == 0)
    return 0;
  else if (n <= 0 || r < 0 || r > n)
    return -INF;
  ld val = 0;
  for (ll i = 0; i < r; i++) {
    val += log(n - i);
    val -= log(i + 1);
  }

  return val;
}

ll bin_search(ll key, ll A[], ll left, ll right) {
  // return the index idx where A[idx] = key.
  // A[left] is start and A[right] is end..
  // In other words, A[right], not A[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (A[mid] == key)
      return mid;
    else if (A[mid] > key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid + 1;
  }
  return -1;
}

/*
ll bin_search_temp(ll left, ll right, callable judge){
    while(right > left){
        // when seeking lower bound
        ll mid = (right + left) / 2;
        if (judge(mid)) right = mid;
        else left = mid + 1;
        // when seeking upper bound
        ll mid = (right + left + 1) / 2;
        if (judge(mid)) left = mid;
        else right = mid - 1;
    }

    return right;
}

trinary_search

// Care the value EPS!!! Compare to the condition
ld left = 0; ld right = p;
while(abs(right - left) > EPS){
    ld left2 = (2 * left + right) / 3.0;
    ld right2 = (left + 2 * right) / 3.0;
    ld f1 = tak_func(left2);
    ld f2 = tak_func(right2);
    if (f1 <= f2) right = right2;
    else if (f2 <= f1) left = left2;
}

*/

ll lower_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] >= key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] < key && key <= A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] < key && key <= A[mid])
      return mid;
    else if (A[mid - 1] >= key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid;
  }

  return -1; // all the elements < key
}

ll inf_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] > key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] <= key && key < A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] <= key && key < A[mid])
      return mid;
    else if (A[mid - 1] > key)
      right = mid - 1;
    else if (A[mid] <= key)
      left = mid;
  }

  return -1; // all the elements <= key
}

ll upper_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] <= key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] <= key && key < A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] <= key && key < A[mid + 1])
      return mid;
    else if (A[mid] > key)
      right = mid;
    else if (A[mid + 1] <= key)
      left = mid + 1;
  }

  return -1; // all the elements > key
}

ll sup_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] < key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] < key && key <= A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] < key && key <= A[mid + 1])
      return mid;
    else if (A[mid] >= key)
      right = mid;
    else if (A[mid + 1] < key)
      left = mid + 1;
  }

  return -1; // all the elements >= key
}

ll bin_search_vector(ll key, vl v, ll left, ll right) {
  // return the index idx where v[idx] = key.
  // v[left] is start and v[right] is end..
  // In other words, v[right], not v[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (v[mid] == key)
      return mid;
    else if (v[mid] > key)
      right = mid - 1;
    else if (v[mid] < key)
      left = mid + 1;
  }
  return -1;
}

ll lower_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] < key <= v[N]
  // N == -1 if all the elements < key
  return lower_bound(all(v), key) - v.begin();
}

ll inf_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] <= key < v[N] <= key + 1
  // N == -1 if all the elements <= key
  return lower_bound(all(v), key + 1) - v.begin();
}

ll upper_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key < v[N + 1]
  // N == -1 if all the elements > key
  return upper_bound(all(v), key) - v.begin(); // (- 1)
}

ll sup_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key - 1 < key <= v[N + 1]
  // N == -1 if all the elements >= key
  return upper_bound(all(v), key - 1) - v.begin() - 1;
}

ll fact(ll n) {
  if (n == 0)
    return 1;
  return n * fact(n - 1);
}

ll fact_mod(ll n, ll mod) {
  if (n == 0)
    return 1;
  return n * fact_mod(n - 1, mod);
}

bool is_prime(ll n) {
  if (n <= 1)
    return false;
  for (ll i = 2; i * i <= n; i++) {
    if (n % i == 0)
      return false;
  }
  return true;
}

ll bool_sum(ll a1, ll a2) {
  if (a1 == 1 || a2 == 1)
    return 1;
  return 0;
}

mll prime_factorization(ll n) {
  ll i = 2;
  mll table;
  while (i * i <= n) {
    while (n % i == 0) {
      table[i]++;
      n /= i;
    }
    i++;
  }
  if (n > 1)
    table[n] = 1;
  return table;
}

vl divisor_table(ll n) {
  vl table;
  ll i = 1;
  while (i * i <= n) {
    if (n % i == 0) {
      table.pb(i);
      if (i * i != n)
        table.pb(n / i);
    }
    i++;
  }

  sort(all(table));

  return table;
}

ll char_to_idx(char c) {
  ll idx = 0;
  Forchar(cc, 'a', 'z') {
    if (c == cc)
      return idx;
    else
      idx++;
  }
}

ll next_combination(ll sub) {
  /*
  ll n, k; ll bit = (1 << k) - 1;
  for (; bit < (1 << n); bit = next_combination(bit)){
      bool ith = bit & (1 << i);
      procedures...
  }
  */
  ll x = sub & -sub, y = sub + x;
  return (((sub & ~y) / x) >> 1) | y;
}

vl z_algorithm(str s) {
  ll n = s.length();
  vl res(n);
  res[0] = n;
  ll i1 = 1, i2 = 0;

  while (i1 < n) {
    while (i1 + i2 < n && s[i2] == s[i1 + i2])
      ++i2;
    res[i1] = i2;
    if (i2 == 0) {
      ++i1;
      continue;
    }
    ll i3 = 1;
    while (i1 + i3 < n && i3 + res[i3] < i2) {
      res[i1 + i3] = res[i3];
      ++i3;
    }
    i1 += i3, i2 -= i3;
  }
  return res;
}

ll big_number_mod(str s, ll mod) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth) % mod;
    val %= mod;
    tenth *= 10;
    tenth %= mod;
    idx++;
  }

  return val;
}

str bin_expression(ll n, ll dig) {
  str s = "";
  rep(i, dig) {
    s += ltos(n % 2);
    n /= 2;
  }
  reverse(all(s));
  return s;
}

ll string_to_ll(str s) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth);
    tenth *= 10;
    idx++;
  }

  return val;
}

str reflected_string(str s) {
  str t, u;
  ll n = s.length();
  t = s;
  reverse(all(t));
  u = substring(t, 0, n - 2) + s + substring(t, 1, n - 1);
  return u;
}

ld distance_between_point_line(point l_begin, point l_end, point p) {
  ll xl1 = l_begin.x, yl1 = l_begin.y;
  ll xl2 = l_end.x, yl2 = l_end.y;
  ll xp = p.x, yp = p.y;
  ll a = yl2 - yl1;
  ll b = -xl2 + xl1;
  ll c = -a * xl2 - b * yl2;
  return abs(ld(a * xp + b * yp + c)) / ld(sqrt(a * a + b * b));
}

bool is_cross(point l1_begin, point l1_end, point l2_begin, point l2_end) {
  ll x1 = l1_begin.x, y1 = l1_begin.y;
  ll x2 = l1_end.x, y2 = l1_end.y;
  ll x3 = l2_begin.x, y3 = l2_begin.y;
  ll x4 = l2_end.x, y4 = l2_end.y;
  ll val1 = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3);
  ll val2 = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4);
  ll val3 = (x3 - x4) * (y1 - y3) + (y3 - y4) * (x3 - x1);
  ll val4 = (x3 - x4) * (y2 - y3) + (y3 - y4) * (x3 - x2);

  return val1 * val2 < 0 && val3 * val4 < 0;
}

ld space_of_triangle(point p1, point p2, point p3) {
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;
  ll v1 = x2 - x1;
  ll u1 = y2 - y1;
  ll v2 = x3 - x1;
  ll u2 = y3 - y1;
  ld s = ld(v1 * u2 - u1 * v2) / ld(2);
  return abs(s);
}

pair<point, ll> center_2det_of_3points(point p1, point p2, point p3) {
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;
  ll c2x_2 = x2 + x1, c2y_2 = y2 + y1;
  ll c3x_2 = x3 + x1, c3y_2 = y3 + y1;
  ll b2y = x2 - x1, b2x = -y2 + y1;
  ll b3y = x3 - x1, b3x = -y3 + y1;
  ll x1_2 = c3x_2 * b3y - c3y_2 * b3x, y1_2 = c2x_2 * b2y - c2y_2 * b2x;
  ll det = -b3y * b2x + b3x * b2y;
  if (det == 0)
    return {{INF, INF}, det};
  ll cx_2det = -b2x * x1_2 + b3x * y1_2, cy_2det = -b2y * x1_2 + b3y * y1_2;
  return {{cx_2det, cy_2det}, det};
}

ll inversion_number(vl a, ll a_max) {
  /*
  Paramters
  ---------
  a: vector<ll>
      All the elements must be non-negative.
      Prefably the elements are compressed to reduce the computational cost.
  a_max: ll
      The maximum value of the vector a or the value bigger than the value
  stated previously.
  */

  BIT bit(a_max + 1);
  ll val = 0;
  rep(i, a.size()) {
    // i is the number of elements that have lower index than a[i].
    // call the number of elements that have lower value than a[i]
    // by subtracting these two, the residual number is the number of elements
    // that have larger value
    val += i - bit.query(a[i] - 1); // cumulative sum from 0 to a[i] - 1
    bit.update(a[i], 1);
  }
  return val;
}

template <typename T> vector<T> compress(vector<T> v) {
  // sort and remove all the duplicated values
  sort(all(v));
  v.erase(unique(all(v)), v.end());
  return v;
}

template <typename T> map<T, ll> dict(const vector<T> &v) {
  map<T, ll> d;
  rep(i, v.size()) d[v[i]] = i;
  return d;
}

/*
const ll N_VERTEX = 310;
ll a, b, t;
ll dist[N_VERTEX][N_VERTEX];

void warshall_floyd(ll n){
    // rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(k, n) rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][k] +
dist[k][j]);
}

int main(void){
    in2(n, m);
    rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        dist[a][b] = t;
        dist[b][a] = t;
    }

    warshall_floyd(n);
}

ll dist[1005], vertex_pre[1005];

void dijkstra(ll start, ll n) {
    priority_queue<pl, vector<pl>, greater<pl>> edge_costs;
    fill(dist, dist + n, INF); // fill(vertex_pre, vertex_pre + n, -1);
    dist[start] = 0;
    edge_costs.push(pl(0, start));

    while (!edge_costs.empty()) {
        pl edge_cost = edge_costs.top();
        edge_costs.pop();
        ll idx = edge_cost.second;
        ll cost = edge_cost.first;
        if (dist[idx] < cost) continue;

        rep(i, graph[idx].size()){
            edge e = graph[idx][i];
            if (dist[e.to] > dist[idx] + e.cost){
                dist[e.to] = dist[idx] + e.cost;
                // vertex_pre[e.to] = idx;
                edge_costs.push(pl(dist[e.to], e.to));
            }
        }
    }
}

vl get_predecessor(ll g){
    vl path;
    for (; g != -1; g = vertex_pre[g]) path.pb(g);

    reverse(all(path));
    return path;
}

int main(void){
    in2(n, m);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        G[a].pb({b, t});
        G[b].pb({a, t});
    }

    dijkstra(0, n);
}


# ABC061D

bool find_negative_cycle(ll goal){
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[e.to] != INF && dist[e.to] > dist[v] + e.cost){
            dist[e.to] = -INF;
            if (goal == -1) return true;
            else if (goal == e.to) return true;
        }
    }
    return false;
}

bool bellman_ford(ll start, ll n, ll goal){
    // if there is a closed circuit, it returns false. (when goal == -1)
    // if the distance to goal cannot be obtained, it returns false (when goal
!= -1) fill(dist, dist + n, INF); dist[start] = 0; rep(i, n) rep(v, n) rep(k,
graph[v].size()){ edge e = graph[v][k]; if (dist[v] != INF && dist[e.to] >
dist[v] + e.cost) dist[e.to] = dist[v] + e.cost;
    }

    if (find_negative_cycle(goal)) return false;
    return true;
}

*/

/*
    # 1. The usage of map pair
    map<pl, ll> cnt;
    cnt[{i, j}] = 0;
    items(kv, cnt){
        prl2(kv.first, kv.second);
    }

    # 2. The usage of next_permutation and combination (factorial search)
    ll a[8];
    rep(i, 8) a[i] = i;
    sort(a, a + 8);
    do{
    }while(next_permutation(a, a+n));

    // here, combination
    ll n, r;
    ll i1, i2, ..., ir;
    For(i1, n - r, n) For(i2, n - r - 1, i1) ... For(ir, n - 2 * r + 1, irr){
   process;}

    # 3. bit search
    ll n;
    in1(n);
    const ll base = 3;
    ll upper = power_normal(base, n);

    rep(i, upper){
        ll tmp = i;
        rep(j, n){
            rep(k, base) if (tmp % base == k) prl(k);
            tmp /= base;
        }
    }

    # 4. imos method
    // used when we would like to count the number which
    // shows how many times the numbers between l and r belongs to smt.
    // This method is composed of three process.
    ll n, m, s[MAX_M], l, r;
    in2(n, m);
    rep(i, m) s[i] = 0;

    // 1st step
    rep(i, n){
        in3(l, r, c);
        l--; r--; // if l starts from 1.
        s[l] += c; s[r + 1] -= c;
    }

    // 2nd step
    rep(i, m - 1) s[i + 1] += s[i];
    // 3rd step: judgement...

    #5. shakutori method (syakutori)
    // 1. strech right side while the condition is met.
    // 2. renew the answer
    // 3. increments left side
    // 4. Back to 1. (l <= r must be satisfied all the time.)

    ll l = 0; ll r = 0;
    while (l < n){
        r = max(r, l);
        if (l == r) r++;
        while(r < n && cond) r++;
        answer += r - l; l++;
    }
    prl(answer);

    #6. priority queue

    pq q;
    ll answer = 0;
    ll v;

    rep(i, n) q.push(a[i]);
    rep(i, m){
        v = q.top(); q.pop(); // get the top value and dump the value from queue
        v /= 2; q.push(v); // add the new value
    }

    while(!q.empty()){
        answer += q.top();
        q.pop();
    }

    #7. The shortest path (distance) between the k-th edge and another edge
   (Tree) ll depth[MAX_N]; gr tree[MAX_N];

    void dfs(ll source, ll parent, ll all_cost){
        depth[source] = all_cost;
        items(e, tree[source]){
            if (e.to == parent) continue;
            dfs(e.to, source, all_cost + e.cost);
        }
    }

    ll n, k, a, b, c;

    in2(n, k);
    rep(i, n - 1){
        in3(a, b, c);
        a--; b--;
        tree[a].pb({b, c});
        tree[b].pb({a, c});
    }

    k--;
    dfs(k, -1, 0);

    #10. Visiting Subtree using recurrent function (ABC138D)

    gr tree[MAX_N];
    ll c[MAX_N];
    bool visited[MAX_N];

    void dfs(ll source, ll parent, ll val){
        visited[source] = true;
        c[source] += val;
        rep(i, tree[source].size()){
            rep(i, m){ll res = n % match[i].e1;}
            ll vertex = tree[source][i].to;
            if (vertex == parent) continue;
            dfs(vertex, source, c[source]);
        }
    }

    #11. bfs ABC146D, ABC007C
    1. first create a tree.
    2. start searching from a node.
    3. do some processes and push nodes connected with a given target node in
   BFS.
    4. repeat a series of procedure until queue is empty.

    queue<pl> q;

    void bfs(ll source, ll parents){
        ll n_edge = G[source].size();
        if (parents != -1) dist[source] = min(dist[source], dist[parents] + 1);
        if (visited[source]) return;
        visited[source] = true;

        rep(idx, n_edge){
            ll target = G[source][idx].to;
            if (target == parents) continue;
            q.push(mp(target, source));
        }
    }

    q.push(mp(sg.e1, -1));
    while(!q.empty()){
        pl source = q.front(); q.pop();
        bfs(source.e1, source.e2);
    }

    #12. grid to distance matrix (dx, dy)
    ll w, h;
    ll pos_to_idx(ll x, ll y){
        return y * w + x;
    }

    pl idx_to_pos(ll idx){
        return mp(idx % w, idx / w);
    }

    int main(void){
        in2(h, w);
        rep(y, h){
            in1(s);
            rep(x, w){
                if (s[x] == '#') wall[x][y] = true;
                else wall[x][y] = false;
            }
        }

        rep(i1, h * w)rep(i2, h * w) dist[i1][i2] = INF * (i1 != i2);

        rep(x, w)rep(y, h){
            ll idx1 = pos_to_idx(x, y); ll idx2;
            if (wall[x][y]) continue;
            if (x != 0 && !wall[x - 1][y]){
                idx2 = pos_to_idx(x - 1, y);
                // if warshall floyd
                dist[idx1][idx2] = 1;
                // if dijkstra
                // graph[idx1].pb({idx2, 1});
            }
            if (x != w - 1 && !wall[x + 1][y]){
                idx2 = pos_to_idx(x + 1, y);
                dist[idx1][idx2] = 1;
                // graph[idx1].pb({idx2, 1});
            }
            if (y != 0 && !wall[x][y - 1]){
                idx2 = pos_to_idx(x, y - 1);
                dist[idx1][idx2] = 1;
                // graph[idx1].pb({idx2, 1});
            }
            if (y != h - 1 && !wall[x][y + 1]){
                idx2 = pos_to_idx(x, y + 1);
                dist[idx1][idx2] = 1;
                // graph[idx1].pb({idx2, 1});
            }
        }
    }


*/

/*
# the operators regarding bit
& (AND), | (OR), ^ (XOR)
- (REVERSE), >> (SMALLER SHIFT)
<< (BIGGER SHIFT)


     x1: 0000 0001 0010 0101 0110 0111 0111
     x2: xxxx 0001 0011 0100 0101 1000 0110
x1 & x2: 0000 0001 0010 0100 0100 0000 0110

      x: 1001 1010  1100 1011 1101  1111
x & - x: 0001 0010  0100 0001 0001  0001
    sum: 1010 1100 10000 1100 1100 10000

x << y is x * 2 ** y
x >> y is rep(i, y) x = x // 2

####### Attention #######
1 << i and 1LL << i is different. If programs show WA, try switch to this one.
Let S be a bit sequence and i be a non-negative integer
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & -(1 << i) -> S - {i}
__builtin_popcount(S) -> the number of elements in S
S = 0 -> S is an empty set
__builtin_popcountl(i) -> the number of 1 in binary
S = (1 << n) - 1 -> S includes all the elements up to the n-th

#Conditional Operator
condition ? true : false;

#iterator
type declaration: auto
value reference: *itr
increment: itr++
decrement: itr--
substitution of value: *itr = smt

# inclusion-exclusion principle
|U[i = 1 to n] Ai| = sum[i = 1 to n] |Ai| - sum[i < j]|Ai ^ Aj| + ... + (-1)^(n
- 1) |^[i = 1 to n]Ai|
*/

const ll MAX_N = 200005;
bool okay = false;
ll answer = 0;
ll n, m, a, b, c, d;

void solve() {
  ll hoge = 2 * m;
  d = (n - 1) / hoge;
  rep(i, m) { prl2(1 + d * i, n - i * d); }
  // check negative MOD
  // check index flow
  // check overwrite of the input variables
}

int main(void) {
  in1(n);
  // assert(n <= 400);
  solve();

  return 0;
}