#include <iostream>
#include <algorithm>
#include <array>
#include <bitset>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <string>
#include <tuple>
#include <list>
#include <unordered_set>
#include <bit>
#include <chrono>
#include <functional>
#include <iomanip>
#include <utility>
#include <type_traits>
#include <cassert>
#include <cctype>
#include <cmath>
#include <cstring>
#include <ctime>
#include <limits>
#include <ranges>
#include <concepts>

#define TP template
#define TN typename
#define TE TP <TN T>
#define TES TP <TN T, TN ...S>
#define Z auto
#define var const Z &
#define ep emplace_back
#define eb emplace
#define fi first
#define se second
#define bg begin
#define ed end
#define all(x) bg(x), ed(x)

#define OV(a, b, c, d, e, ...) e
#define FO4(i, a, b, c) for (int i = (a); i < (b); i += (c))
#define FO3(i, a, b) FO4(i, a, b, 1)
#define FO2(i, a) FO3(i, 0, a)
#define FO1(a) FO2(_, a)
#define FOR(...) OV(__VA_ARGS__, FO4, FO3, FO2, FO1)(__VA_ARGS__)
#define FF4(i, a, b, c) for (int i = (b) - 1; i >= (a); i -= (c))
#define FF3(i, a, b) FF4(i, a, b, 1)
#define FF2(i, a) FF3(i, 0, a)
#define FF1(a) FF2(_, a)
#define FOR_R(...) OV(__VA_ARGS__, FF4, FF3, FF2, FF1)(__VA_ARGS__)
#define FOR_subset(t, s) for (int t = (s); t > -1; t = (t == 0 ? -1 : (t - 1) & s))

#define sort ranges::sort

using namespace std;

TE using vc = vector<T>;
TE using vvc = vc<vc<T>>;
TE using T1 = tuple<T>;
TE using T2 = tuple<T, T>;
TE using T3 = tuple<T, T, T>;
TE using T4 = tuple<T, T, T, T>;
TE using max_heap = priority_queue<T>;
TE using min_heap = priority_queue<T, vc<T>, greater<T>>;
using u8 = unsigned char;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using ld = long double;
using i128 = __int128;
using u128 = __uint128_t;
using f128 = __float128;
using u16 = uint16_t;
using PII = pair<int, int>;
using PLL = pair<ll, ll>;
using vi = vc<int>;
using vl = vc<ll>;

#ifdef YRSD
constexpr bool dbg = 1;
#else
constexpr bool dbg = 0;

#endif

TES concept nof = (not same_as<T, S> and ...);

TE constexpr bool can_in = 0;
TE constexpr bool can_out = 0;
TP<> constexpr bool can_in<istream> = 1;
TP<> constexpr bool can_out<ostream> = 1;

istream &operator>>(istream &I, i128 &x) {
  static string s;
  I >> s;
  int f = s[0] == '-';
  x = 0;
  const int N = (int)s.size();
  FOR(i, f, N) x = x * 10 + s[i] - '0';
  if (f) x = -x;
  return I;
}
ostream &operator<<(ostream &O, i128 x) {
  static string s;
  s.clear();
  bool f = x < 0;
  if (f) x = -x;
  while (x) s += '0' + x % 10, x /= 10;
  if (s.empty()) s += '0';
  if (f) s += '-';
  reverse(all(s));
  return O << s;
}

istream &operator>>(istream &I, f128 &x) {
  static string s;
  I >> s, x = stold(s);
  return I;
}
ostream &operator<<(ostream &O, const f128 x) { return O << ld(x); }

TP <TN I, TN... S> requires(can_in<I>)
I &operator>>(I &in, tuple<S...> &a) {
  apply([&in](Z &...s) { ((in >> s), ...); }, a);
  return in;
}

TP <TN I, TN A, TN B> requires(can_in<I>)
I &operator>>(I &in, pair<A, B> &x) {
  return in >> x.fi >> x.se;
}

TP <TN U, TN A, TN B> requires(can_out<U>)
U &operator<<(U &out, const pair<A, B> &x) {
  return out << x.fi << ' ' << x.se;
}

TP <TN I, TN T> requires(can_in<I>)
I &operator>>(I &in, vc<T> &a) {
  for (Z &x : a) in >> x;
  return in;
}

TP <TN U, TN T> requires(can_out<U>)
U &operator<<(U &out, const vc<T> &a) {
  if (a.empty()) return out;
  Z i = bg(a);
  out << *i++;
  for (; i != ed(a); ++i) out << ' ' << *i;
  return out;
}

TP <TN I, TN T, std::size_t N>  requires(can_in<I>)
I &operator>>(I &in, array<T, N> &a) {
  FOR(i, N) in >> a[i];
  return in;
}

TP <TN U, TN T, std::size_t N> requires(can_out<U>)
U &operator<<(U &out, const array<T, N> &a) {
  out << a[0];
  FOR(i, 1, N) out << ' ' << a[i];
  return out;
}

void IN() {}
TE void IN(T &x, Z &...s) { cin >> x, IN(s...); }
void print() { cout << '\n'; }
TES void print(T &&x, S &&...y) {
  cout << x;
  if constexpr (sizeof...(S)) cout << ' ';
  print(forward<S>(y)...);
}
void put() {}
TES void put(T &&x, S &&...y) {
  cout << x;
  put(forward<S>(y)...);
}

#define INT(...)  int    __VA_ARGS__; IN(__VA_ARGS__)
#define UINT(...) uint   __VA_ARGS__; IN(__VA_ARGS__)
#define LL(...)   ll     __VA_ARGS__; IN(__VA_ARGS__)
#define ULL(...)  ull    __VA_ARGS__; IN(__VA_ARGS__)
#define I128(...) i128   __VA_ARGS__; IN(__VA_ARGS__)
#define STR(...)  string __VA_ARGS__; IN(__VA_ARGS__)
#define CH(...)   char   __VA_ARGS__; IN(__VA_ARGS__)
#define REAL(...) re     __VA_ARGS__; IN(__VA_ARGS__)
#define VEC(T, a, n) vc<T> a(n); IN(a)

void YES(bool o = 1) { print(o ? "YES" : "NO"); }
void Yes(bool o = 1) { print(o ? "Yes" : "No"); }
void yes(bool o = 1) { print(o ? "yes" : "no"); }
void NO(bool o = 1) { YES(not o); }
void No(bool o = 1) { Yes(not o); }
void no(bool o = 1) { yes(not o); }
void ALICE(bool o = 1) { print(o ? "ALICE" : "BOB"); }
void Alice(bool o = 1) { print(o ? "Alice" : "Bob"); }
void alice(bool o = 1) { print(o ? "alice" : "bob"); }
void BOB(bool o = 1) { ALICE(not o); }
void Bob(bool o = 1) { Alice(not o); }
void bob(bool o = 1) { alice(not o); }
void POSSIBLE(bool o = 1) { print(o ? "POSSIBLE" : "IMPOSSIBLE"); }
void Possible(bool o = 1) { print(o ? "Possible" : "Impossible"); }
void possible(bool o = 1) { print(o ? "possible" : "impossible"); }
void IMPOSSIBLE(bool o = 1) { POSSIBLE(not o); }
void Impossible(bool o = 1) { Possible(not o); }
void impossible(bool o = 1) { possible(not o); }
void TAK(bool o = 1) { print(o ? "TAK" : "NIE"); }
void NIE(bool o = 1) { TAK(not o); }

#if (__cplusplus >= 202002L)
#include <numbers>
constexpr ld pi = numbers::pi_v<ld>;
#endif

TE constexpr T inf = numeric_limits<T>::max();
template <> constexpr i128 inf<i128> = i128(inf<ll>) * 2'000'000'000'000'000'000;
template <typename T, typename U>
constexpr pair<T, U> inf<pair<T, U>> = {inf<T>, inf<U>};

TE constexpr static inline int pc(T x) { return popcount(make_unsigned_t<T>(x)); }
constexpr static inline ll si(var a) { return a.size(); }

void reverse(Z &a) { reverse(all(a)); }

void unique(Z &a) {
  sort(a);
  a.erase(unique(all(a)), ed(a));
}
TE vc<int> inverse(const vc<T> &a) {
  int N = si(a);
  vc<int> b(N, -1);
  FOR(i, N) if (a[i] != -1) b[a[i]] = i;
  return b;
}

Z QMAX(var a) { return *max_element(all(a)); }
Z QMIN(var a) { return *min_element(all(a)); }
TE Z QMAX(T l, T r) { return *max_element(l, r); }
TE Z QMIN(T l, T r) { return *min_element(l, r); }
constexpr bool chmax(Z &a, var b) { return (a < b ? a = b, 1 : 0); }
constexpr bool chmin(Z &a, var b) { return (a > b ? a = b, 1 : 0); }

vc<int> argsort(var a) {
  vc<int> I(si(a));
  iota(all(I), 0);
  sort(I, [&](int i, int k) { return a[i] < a[k] or (a[i] == a[k] and i < k); });
  return I;
}
TE vc<T> rearrange(const vc<T> &a, const vc<int> &I) {
  int N = si(I);
  vc<T> b(N);
  FOR(i, N) b[i] = a[I[i]];
  return b;
}
template <int of = 1, typename T>
vc<T> pre_sum(const vc<T> &a) {
  int N = si(a);
  vc<T> c(N + 1);
  FOR(i, N) c[i + 1] = c[i] + a[i];
  if (of == 0) c.erase(bg(c));
  return c;
}

TE constexpr static int topbit(T x) {
  if (x == 0) return -1;
  if constexpr (sizeof(T) <= 4) return 31 - __builtin_clz(x);
  else return 63 - __builtin_clzll(x);
}
TE constexpr static int lowbit(T x) {
  if (x == 0) return -1;
  if constexpr (sizeof(T) <= 4) return __builtin_ctz(x);
  else return __builtin_ctzll(x);
}

TE constexpr inline T floor(T x, T y) { return x / y - (x % y and (x ^ y) < 0); }
TE constexpr inline T ceil(T x, T y) { return floor(x + y - 1, y); }
TE constexpr inline T bmod(T x, T y) { return x - floor(x, y) * y; }
TE constexpr inline pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return pair{q, x - q * y};
}

TE T SUM(var v) { return accumulate(all(v), T()); }
TE T SUM(Z l, Z r) { return accumulate(l, r, T()); }
int lb(var a, Z x) { return lower_bound(all(a), x) - a.begin(); }
TE int lb(T l, T r, Z x) { return lower_bound(l, r, x) - l; }
int ub(var a, Z x) { return upper_bound(all(a), x) - a.begin(); }
TE int ub(T l, T r, Z x) { return upper_bound(l, r, x) - l; }

template <bool ck = 1>
ll bina(Z f, ll l, ll r) {
  if (ck) assert(f(l));
  while (abs(l - r) > 1) {
    ll x = (r + l) >> 1;
    (f(x) ? l : r) = x;
  }
  return l;
}
TE T bina_real(Z f, T l, T r, int c = 100) {
  while (c--) {
    T x = (l + r) / 2;
    (f(x) ? l : r) = x;
  }
  return (l + r) / 2;
}

TE T pop(vc<T> &a) {
  T x = a.back();
  a.pop_back();
  return x;
}
TE T pop(max_heap<T> &q) {
  T x = q.top();
  q.pop();
  return x;
}
TE T pop(min_heap<T> &q) {
  T x = q.top();
  q.pop();
  return x;
}
char pop(string &s) {
  char x = s.back();
  s.pop_back();
  return x;
}

void setp(int x) { cout << fixed << setprecision(x); }

TE inline void sh(vc<T> &a, int N, T b = {}) { a.resize(N, b); }

namespace fio {
  static constexpr uint sz = 1 << 17;
  char a[sz];
  char b[sz];
  char t[100];

  uint l = 0, r = 0, pr = 0;

  struct Pre {
    char num[10000][4];
    constexpr Pre() : num() {
      for (int i = 0; i < 10000; i++) {
        int n = i;
        for (int j = 3; j >= 0; j--) {
          num[i][j] = n % 10 | '0';
          n /= 10;
        }
      }
    }
  } constexpr pre;

  inline void load() {
    memcpy(a, a + l, r - l);
    r = r - l + fread(a + r - l, 1, sz - r + l, stdin);
    l = 0;
    if (r < sz) a[r++] = '\n';
  }

  inline void flush() { fwrite(b, 1, pr, stdout), pr = 0; }

  inline void rd(char &c) {
    do {
      if (l + 1 > r) load();
      c = a[l++];
    } while (isspace(c));
  }

  inline void rd(string &s) {
    s.clear();
    char c;
    do {
      if (l + 1 > r) load();
      c = a[l++];
    } while (isspace(c));
    do {
      s += c;
      if (l == r) load();
      c = a[l++];
    } while (!isspace(c));
  }

  TE inline void rd_re(T &x) {
    static string s;
    rd(s);
    x = stod(s);
  }

  TE inline void rd_inte(T &x) {
    if (l + 100 > r) load();
    char c;
    do c = a[l++];
    while (c < '-');
    bool op = 0;
    if constexpr (is_signed_v<T> or is_same_v<T, i128>) {
      if (c == '-') op = 1, c = a[l++];
    }
    x = 0;
    while ('0' <= c) x = x * 10 + (c & 15), c = a[l++];
    if constexpr (is_signed_v<T> or is_same_v<T, i128>) {
      if (op) x = -x;
    }
  }

  struct Fin {
    inline Fin &operator>>(char &c) {
      rd(c);
      return *this;
    }

    inline Fin &operator>>(string &s) {
      rd(s);
      return *this;
    }

    TE requires(is_integral_v<T> or is_same_v<T, i128> or is_same_v<T, u128>)
    inline Fin &operator>>(T &x) {
      rd_inte(x);
      return *this;
    }

    TE requires(is_floating_point_v<T> or is_same_v<T, f128>)
    inline Fin &operator>>(T &x){
      rd_re(x);
      return *this;
    }
  } fin;

  inline void wt(char c) {
    if (pr == sz) flush();
    b[pr++] = c;
  }

  inline void wt(const string &s) {
    for (char c : s) wt(c);
  }

  inline void wt(const char *s) {
    size_t si = strlen(s);
    for (size_t i = 0; i < si; i++) wt(s[i]);
  }

  TE inline void wt_inte(T x) {
    if (pr > sz - 100) flush();
    if (x < 0) b[pr++] = '-', x = -x;
    int outi = 96;
    for (; x >= 10000; outi -= 4) {
      memcpy(t + outi, pre.num[x % 10000], 4);
      x /= 10000;
    }
    if (x >= 1000) {
      memcpy(b + pr, pre.num[x], 4);
      pr += 4;
    } else if (x >= 100) {
      memcpy(b + pr, pre.num[x] + 1, 3);
      pr += 3;
    } else if (x >= 10) {
      int q = (x * 103) >> 10;
      b[pr] = q | '0';
      b[pr + 1] = (x - q * 10) | '0';
      pr += 2;
    } else b[pr++] = x | '0';
    memcpy(b + pr, t + outi + 4, 96 - outi);
    pr += 96 - outi;
  }

  int w = 10;
  TE inline void wt_real(T x) {
    ostringstream oss;
    oss << fixed << setprecision(w) << double(x);
    string s = oss.str();
    wt(s);
  }

  struct Fout {
    void setp(int x) { w = x; }

    inline Fout &operator<<(const char &c) {
      wt(c);
      return *this;
    }

    inline Fout &operator<<(const string &s) {
      wt(s);
      return *this;
    }

    TE requires(is_integral_v<T> or is_same_v<T, i128> or is_same_v<T, u128>)
    inline Fout &operator<<(const T &x) {
      wt_inte(x);
      return *this;
    }

    TE requires(is_floating_point_v<T> or is_same_v<T, f128>)
    inline Fout &operator<<(const T &x) {
      wt_real(x);
      return *this;
    }
  } fout;

  inline void __attribute__((destructor)) _d() { flush(); }

  inline void sc() {}
  TE inline void sc(T &x, Z &...s) { fin >> x, sc(s...); }

  void inline pt() { fout << '\n'; }
  TES inline void pt(T &&x, S &&...y) {
    fout << x;
    if constexpr (sizeof...(S)) fout << ' ';
    pt(forward<S>(y)...);
  }

  void inline pu() {}
  TES inline void pu(T &&x, S &&...y) {
    fout << x;
    pu(forward<S>(y)...);
  }
}
template <>
constexpr bool can_in<fio::Fin> = 1;
template <>
constexpr bool can_out<fio::Fout> = 1;

#define setp(x) fio::fout.setp(x)
#define IN fio::sc
#define print fio::pt
#define put fio::pu

struct dsu {
  int c;
  vc<int> fa;
  dsu(int N = 0) : c(N), fa(N, -1) {}

  int f(int x) {
    while (fa[x] >= 0) {
      int p = fa[fa[x]];
      if (p < 0) return fa[x];
      x = fa[x] = p;
    }
    return x;
  }

  int operator[](int x) { return f(x); }

  bool merge(int x, int y) {
    x = f(x), y = f(y);
    if (x == y) return 0;
    if (fa[x] > fa[y]) swap(x, y);
    fa[x] += fa[y];
    fa[y] = x;
    --c;
    return 1;
  }

  bool set(int x, int y) {
    x = f(x), y = f(y);
    if (x == y) return 0;
    fa[x] += fa[y];
    fa[y] = x;
    --c;
    return 1;
  }

  int size(int x) { return -fa[f(x)]; }

  int count() const { return c; }

  bool same(int x, int y) { return f(x) == f(y); }

  void build(int N) { fa.assign(N, -1), c = N; }

  void reset() { fill(all(fa), -1), c = si(fa); }

  vc<vc<int>> group() {
    int N = si(fa);
    vc<vc<int>> v(N), s;
    FOR(i, N) v[f(i)].ep(i);
    FOR(i, N) if (not v[i].empty()) s.ep(v[i]);
    return s;
  }

  void pr() {
    int N = si(fa);
    vc<int> res(N);
    FOR(i, N) res[i] = f(i);
    print("fa:", res);
  }
};

TE struct hashmap {
  uint ls, ms;
  vc<ull> a;
  vc<T> b;
  vc<u8> vis;

  ull hash(ull x) const {
    static const ull bs =
        chrono::steady_clock::now().time_since_epoch().count();
    x += bs;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return (x ^ (x >> 31)) & ms;
  }

  void extend() {
    vc<pair<ull, T>> s;
    int N = si(vis);
    s.reserve(N / 2 - ls);
    FOR(i, N) if (vis[i]) s.ep(a[i], b[i]);
    build(si(s) << 1);
    for (var [l, r] : s) (*this)[l] = r;
  }

  hashmap(uint N = 0) { build(N); }

  void build(uint N) {
    uint k = 8;
    while (k < (N << 1)) k <<= 1;
    ls = k >> 1, ms = k - 1;
    a.resize(k);
    b.resize(k);
    vis.assign(k, 0);
  }

  void clear() {
    fill(all(vis), 0);
    ls = (ms + 1) >> 1;
  }

  ll size() const { return vis.size() / 2 - ls; }

  int id(ull k) const {
    int i = hash(k);
    while (vis[i] and a[i] != k) i = (i + 1) & ms;
    return i;
  }

  T &operator[](ull k) {
    if (ls == 0) extend();
    int i = id(k);
    if (not vis[i]) {
      vis[i] = 1;
      a[i] = k;
      b[i] = T();
      --ls;
    }
    return b[i];
  }

  T get(ull k, T fl) const {
    int i = id(k);
    return (vis[i] ? b[i] : fl);
  }

  bool contains(ull k) const {
    int i = id(k);
    return vis[i] and a[i] == k;
  }

  vc<pair<ull, T>> get_all() const {
    int N = si(vis);
    vc<pair<ull, T>> s;
    FOR(i, N) if (vis[i]) s.ep(a[i], b[i]);
    return s;
  }
};

TE struct edge {
  int f, to;
  T w;
  int id;
  operator int() const { return to; }
};
template <typename T = int, bool dir = 0>
struct graph {
  static constexpr bool is_dir = dir;
  int N, M;
  using cost_type = T;
  using ee = edge<T>;
  vc<ee> es;
  vc<int> in;
  vc<ee> c;

  bool ok;

  bool isok() { return ok; }

  struct px {
    const graph *g;
    int l, r;
    px(const graph *g, int l, int r) : g(g), l(l), r(r) {}
    const ee *begin() const {
      if (l == r) return 0;
      return &g->c[l];
    }
    const ee *end() const {
      if (l == r) return 0;
      return &g->c[r];
    }
  };

  px operator[](int i) const {
    assert(ok);
    return {this, in[i], in[i + 1]};
  }

  graph() : N(0), M(0), ok(0) {}
  graph(int N) : N(N), M(0), ok(0) {}

  void add(int f, int t, T w = 1, int i = -1) {
    assert(not ok);
    assert(-1 < f and -1 < t and t < N and f < N);
    if (i == -1) i = M;
    es.ep(ee{f, t, w, i});
    ++M;
  }

  void build() {
    assert(not ok);
    ok = 1;
    in.assign(N + 1, 0);
    for (Z &&e : es) {
      in[e.f + 1]++;
      if (not dir) in[e.to + 1]++;
    }
    FOR(i, N) in[i + 1] += in[i];
    Z cc = in;
    c.resize(in.back() + 1);
    for (Z &&e : es) {
      c[cc[e.f]++] = e;
      if (not dir) c[cc[e.to]++] = {e.to, e.f, e.w, e.id};
    }
  }

  template <bool wt = 0, int of = 1>
  void sc() { sc<wt, of>(N - 1); }
  template <bool wt = 0, int of = 1>
  void sc(int M) {
    es.reserve(M * (dir ? 1 : 2));
    FOR(M) {
      INT(x, y);
      x -= of, y -= of;
      if (not wt) {
        add(x, y);
      } else {
        T w;
        IN(w);
        add(x, y, w);
      }
    }
    build();
  }

  vc<int> deg() {
    vc<int> a(N);
    for (Z &&e : es) ++a[e.f], ++a[e.to];
    return a;
  }

  pair<vc<int>, vc<int>> deg_inout() {
    vc<int> a(N), b(N);
    for (Z &&e : es) ++a[e.to], ++b[e.f];
    return {a, b};
  }

  vc<int> ni;
  vc<u8> vis;

  graph<T, dir> rearrange(const vc<int> &v, bool keep_eid = 0) {
    if (si(ni) != N) ni.assign(N, -1);
    int N = si(v);
    FOR(i, N) ni[v[i]] = i;
    graph<T, dir> g(N);
    vc<int> s;
    FOR(i, N) {
      for (Z &&e : (*this)[v[i]]) {
        if (si(vis) <= e.id) vis.resize(e.id + 1);
        if (vis[e.id]) continue;
        int f = e.f, to = e.to;
        if (ni[f] != -1 and ni[to] != -1) {
          s.ep(e.id);
          vis[e.id] = 1;
          int id = (keep_eid ? e.id : -1);
          g.add(ni[f], ni[to], e.w, id);
        }
      }
    }
    FOR(i, N) ni[v[i]] = -1;
    for (int i : s) vis[i] = 0;
    return g.build(), g;
  }

  ull has(ull x, ull y) {
    if (not dir and x > y) swap(x, y);
    return x * N + y;
  }

  hashmap<int> mp;
  int get_eid(ull x, ull y) {
    if (mp.size() == 0) {
      mp.build(N - 1);
      for (Z &&e : es) {
        ull x = e.f, y = e.to;
        ull k = has(x, y);
        mp[k] = e.id;
      }
    }
    return mp.get(has(x, y), -1);
  }

  graph rev() const requires(dir) {
    graph ng(N);
    for (Z &&[f, t, w, id] : es) ng.add(t, f, w, id);
    return ng;
  }
};

TE struct hld {
  using G = graph<T, 0>;
  G &g;
  int N, t = 0;
  vc<int> L, R, hd, V, fa, to, d;

  hld(G &g, int r = 0)
      : g(g), N(g.N), L(N, -1), R(L), hd(N, r), V(L), fa(L), to(L), d(N) {
    if (r == -1) return;
    assert(g.isok());
    dfs(r, -1);
    hl(r, r);
  }

  void dfs(int n, int f) {
    fa[n] = f;
    R[n] = 1;
    int l = g.in[n], r = g.in[n + 1], m = 0;
    Z &c = g.c;
    if (r - l > 1 and c[l].to == f) swap(c[l], c[l + 1]);
    FOR(i, l, r) if (c[i].to != f) {
      Z e = c[i];
      to[e.to] = e.id;
      d[e.to] = d[n] + 1;
      dfs(e.to, n);
      R[n] += R[e.to];
      if (chmax(m, R[e.to]) and l < i) swap(c[l], c[i]);
    }
  }

  void hl(int n, int p) {
    R[n] += L[n] = t;
    V[t++] = n;
    bool f = 1;
    for (Z &&e : g[n]) if (e.to != p) {
      hd[e.to] = f ? hd[n] : e.to;
      f = 0;
      hl(e.to, n);
    }
  }

  vc<int> hp(int n) {
    vc<int> s{n};
    while (1) {
      int x = hc(s.back());
      if (x == -1 or hd[x] != n) return s;
      s.ep(x);
    }
  }

  inline int hc(int x) {
    int i = L[x] + 1;
    if (i == N) return -1;
    int a = V[i];
    return fa[a] == x ? a : -1;
  }

  int ev(int i) {
    Z &e = g.es[i];
    return (fa[e.f] == e.to ? e.f : e.to);
  }

  int ve(int x) { return to[x]; }

  int gei(int x, int y) {
    if (fa[x] != y) swap(x, y);
    assert(fa[x] == y);
    return to[x];
  }

  int el(int i) { return 2 * L[i] - d[i]; }
  int er(int i) { return 2 * R[i] - d[i] - 1; }

  int la(int n, int k) {
    assert(k <= d[n]);
    while (1) {
      int x = hd[n];
      if (L[n] - k >= L[x]) return V[L[n] - k];
      k -= L[n] - L[x] + 1;
      n = fa[x];
    }
  }

  int lca(int x, int y) {
    for (;; y = fa[hd[y]]) {
      if (L[x] > L[y]) swap(x, y);
      if (hd[x] == hd[y]) return x;
    }
  }

  int dist(int a, int b) { return d[a] + d[b] - 2 * d[lca(a, b)]; }

  int meet(int a, int b, int c) { return lca(a, b) ^ lca(a, c) ^ lca(b, c); }

  bool ins(int x, int y) { return L[y] <= L[x] and L[x] < R[y]; }

  int jump(int x, int y, int k) {
    if (k == 1) {
      if (x == y) return -1;
      return ins(y, x) ? la(y, d[y] - d[x] - 1) : fa[x];
    }
    int c = lca(x, y), a = d[x] - d[c], b = d[y] - d[c];
    if (k > a + b) return -1;
    if (k <= a) return la(x, k);
    return la(y, a + b - k);
  }

  int size(int x, int r = -1) {
    if (r == -1) return R[x] - L[x];
    if (x == r) return N;
    int y = jump(x, r, 1);
    if (ins(x, y)) return R[x] - L[x];
    return N - R[y] + L[y];
  }

  vc<int> size_arr(int r = -1) {
    vc<int> sz(N);
    FOR(i, N) sz[i] = size(i, r);
    return sz;
  }

  vc<int> cc(int n) {
    static vc<int> s;
    s.clear();
    for (Z &&e : g[n]) if (e.to != fa[n]) s.ep(e.to);
    return s;
  }

  vc<int> cl(int n) {
    static vc<int> s;
    s.clear();
    bool f = 1;
    for (Z &&e : g[n]) {
      if (e.to != fa[n]) {
        if (not f) s.ep(e.to);
        f = 0;
      }
    }
    return s;
  }

  vc<PII> dec(int x, int y, bool e) {
    vc<PII> a, b;
    while (1) {
      if (hd[x] == hd[y]) break;
      if (L[x] < L[y]) {
        b.ep(L[hd[y]], L[y]);
        y = fa[hd[y]];
      } else {
        a.ep(L[x], L[hd[x]]);
        x = fa[hd[x]];
      }
    }
    if (L[x] < L[y]) b.ep(L[x] + e, L[y]);
    else if (L[y] + e <= L[x]) a.ep(L[x], L[y] + e);
    reverse(b);
    a.insert(a.end(), all(b));
    return a;
  }

  vc<int> rest_path(int x, int y) {
    vc<int> s;
    for (Z [a, b] : dec(x, y, 0)) {
      if (a <= b) FOR(i, a, b + 1) s.ep(V[i]);
      else FOR_R(i, b, a + 1) s.ep(V[i]);
    }
    return s;
  }

  PII cross(int a, int b, int c, int d) {
    int ab = lca(a, b), ac = lca(a, c), ad = lca(a, d);
    int bc = lca(b, c), bd = lca(b, d), cd = lca(c, d);
    int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd;

    if (x != y) return {x, y};
    int z = ac ^ ad ^ cd;
    if (x != z) x = -1;
    return {x, x};
  }

  int max_path(Z f, int x, int y) {
    if (not f(x)) return -1;
    for (Z [a, b] : dec(x, y, 0)) {
      if (not f(V[a])) return x;
      if (f(V[b])) {
        x = V[b];
        continue;
      }
      int c = bina<0>([&](int c) -> bool { return f(V[c]); }, a, b);
      return V[c];
    }
    return x;
  }
};

struct func_g {
  int N;
  vc<int> to;
  graph<int> g;
  vc<int> rs;
  hld<int> v;

  void gen() {
    dsu f(N);
    FOR(i, N) if (not f.merge(i, to[i])) rs[i] = i;
    FOR(i, N) if (rs[i] == i) rs[f[i]] = i;
    FOR(i, N) rs[i] = rs[f[i]];
    FOR(i, N) {
      if (rs[i] == i) g.add(N, i);
      else g.add(to[i], i);
    }
    g.build();
  }

  func_g(const vc<int> &to)
      : N(si(to)), to(to), g(N + 1), rs(N, -1), v(g, (gen(), N)) {}

  int dist(int a, int b) {
    if (v.ins(a, b)) return v.d[a] - v.d[b];
    int r = rs[a], t = to[r];
    if (v.ins(t, b)) return v.d[a] - v.d[r] + 1 + v.d[t] - v.d[b];
    return inf<int>;
  }

  int jump(int n, ll k) {
    int d = v.d[n];
    if (k < d) return v.jump(n, N, k);
    n = rs[n];
    k -= d - 1;
    int t = to[n], c = v.d[t];
    k %= c;
    if (k == 0) return n;
    return v.jump(t, N, k - 1);
  }

  vc<int> jump_all(ll k) {
    vc<int> res(N, -1);
    vc<vc<PII>> q(N);
    FOR(n, N) {
      int d = v.d[n], r = rs[n];
      if (d - 1 > k) q[n].ep(n, k);
      if (d - 1 <= k) {
        ll kk = k - d + 1;
        int t = to[r], c = v.d[t];
        kk %= c;
        if (kk == 0) {
          res[n] = r;
          continue;
        }
        q[t].ep(n, kk - 1);
      }
    }
    vc<int> pa;
    Z f = [&](Z &f, int n, int p) -> void {
      pa.ep(n);
      for (var [w, k] : q[n]) res[w] = ed(pa)[-1 - k];
      for (int x : g[n]) if (x != p) f(f, x, n);
      pop(pa);
    };
    for (int x : g[N]) f(f, x, N);
    return res;
  }

  bool inc(int n) {
    int r = rs[n];
    if (to[r] == r) return n == r;
    return v.ins(to[r], n);
  }

  vc<int> cc(int n) {
    assert(n == rs[n]);
    vc<int> s{to[n]};
    while (s.back() != n) s.ep(to[s.back()]);
    return s;
  }

  int meet_time(int i, int k) {
    if (i == k) return 0;
    if (rs[i] != rs[k]) return -1;
    int r = rs[i], b = to[r], n = v.d[b] - v.d[r] + 1;
    if ((v.d[i] - v.d[k]) % n != 0) return -1;
    if (v.d[i] == v.d[k]) return v.d[i] - v.d[v.lca(i, k)];
    int x = v.d[i] - v.d[v.lca(b, i)];
    int y = v.d[k] - v.d[v.lca(b, k)];
    return max(x, y);
  }
};

void Yorisou() {
  INT(N, K);
  VEC(ll, a, N);
  a.insert(ed(a), all(a));
  pop(a);
  print(bina([&](ll lm) -> bool {
    int r = 0;
    ll s = 0;
    vc<int> to(N + N + 1, N + N);
    FOR(l, N + N - 1) {
      while (r - l < N and r < N + N - 1 and s < lm) s += a[r++];
      if (s >= lm) to[l] = r;
      s -= a[l];
    }
    to = func_g(to).jump_all(K);
    FOR(i, N) if (to[i] - i <= N) return 1;
    return 0;
  }, 0, SUM<ll>(a) / K + 1));
}

int main() {
  Yorisou();
  return 0;
}