#include<bits/stdc++.h>

#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

  namespace internal {

    // @param n `0 <= n`
    // @return minimum non-negative `x` s.t. `n <= 2**x`
    int ceil_pow2(int n) {
      int x = 0;
      while ((1U << x) < (unsigned int)(n)) x++;
      return x;
    }

    // @param n `1 <= n`
    // @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
    int bsf(unsigned int n) {
#ifdef _MSC_VER
      unsigned long index;
      _BitScanForward(&index, n);
      return index;
#else
      return __builtin_ctz(n);
#endif
    }

  }  // namespace internal

}  // namespace atcoder



#include <utility>

namespace atcoder {

  namespace internal {

    // @param m `1 <= m`
    // @return x mod m
    constexpr long long safe_mod(long long x, long long m) {
      x %= m;
      if (x < 0) x += m;
      return x;
    }

    // Fast moduler by barrett reduction
    // Reference: https://en.wikipedia.org/wiki/Barrett_reduction
    // NOTE: reconsider after Ice Lake
    struct barrett {
      unsigned int _m;
      unsigned long long im;

      // @param m `1 <= m`
      barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

      // @return m
      unsigned int umod() const { return _m; }

      // @param a `0 <= a < m`
      // @param b `0 <= b < m`
      // @return `a * b % m`
      unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
	  (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
      }
    };

    // @param n `0 <= n`
    // @param m `1 <= m`
    // @return `(x ** n) % m`
    constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
      if (m == 1) return 0;
      unsigned int _m = (unsigned int)(m);
      unsigned long long r = 1;
      unsigned long long y = safe_mod(x, m);
      while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
      }
      return r;
    }

    // Reference:
    // M. Forisek and J. Jancina,
    // Fast Primality Testing for Integers That Fit into a Machine Word
    // @param n `0 <= n`
    constexpr bool is_prime_constexpr(int n) {
      if (n <= 1) return false;
      if (n == 2 || n == 7 || n == 61) return true;
      if (n % 2 == 0) return false;
      long long d = n - 1;
      while (d % 2 == 0) d /= 2;
      for (long long a : {2, 7, 61}) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
	  y = y * y % n;
	  t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
	  return false;
        }
      }
      return true;
    }
    template <int n> constexpr bool is_prime = is_prime_constexpr(n);

    // @param b `1 <= b`
    // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
    constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
      a = safe_mod(a, b);
      if (a == 0) return {b, 0};

      // Contracts:
      // [1] s - m0 * a = 0 (mod b)
      // [2] t - m1 * a = 0 (mod b)
      // [3] s * |m1| + t * |m0| <= b
      long long s = b, t = a;
      long long m0 = 0, m1 = 1;

      while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
      }
      // by [3]: |m0| <= b/g
      // by g != b: |m0| < b/g
      if (m0 < 0) m0 += b / s;
      return {s, m0};
    }

    // Compile time primitive root
    // @param m must be prime
    // @return primitive root (and minimum in now)
    constexpr int primitive_root_constexpr(int m) {
      if (m == 2) return 1;
      if (m == 167772161) return 3;
      if (m == 469762049) return 3;
      if (m == 754974721) return 11;
      if (m == 998244353) return 3;
      int divs[20] = {};
      divs[0] = 2;
      int cnt = 1;
      int x = (m - 1) / 2;
      while (x % 2 == 0) x /= 2;
      for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
	  divs[cnt++] = i;
	  while (x % i == 0) {
	    x /= i;
	  }
        }
      }
      if (x > 1) {
        divs[cnt++] = x;
      }
      for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
	  if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
	    ok = false;
	    break;
	  }
        }
        if (ok) return g;
      }
    }
    template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

  }  // namespace internal

}  // namespace atcoder


#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

  namespace internal {

#ifndef _MSC_VER
    template <class T>
    using is_signed_int128 =
      typename std::conditional<std::is_same<T, __int128_t>::value ||
				std::is_same<T, __int128>::value,
				std::true_type,
				std::false_type>::type;

    template <class T>
    using is_unsigned_int128 =
      typename std::conditional<std::is_same<T, __uint128_t>::value ||
				std::is_same<T, unsigned __int128>::value,
				std::true_type,
				std::false_type>::type;

    template <class T>
    using make_unsigned_int128 =
      typename std::conditional<std::is_same<T, __int128_t>::value,
				__uint128_t,
				unsigned __int128>;

    template <class T>
    using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
						  std::true_type,
						  std::false_type>::type;

    template <class T>
    using is_signed_int = typename std::conditional<(is_integral<T>::value &&
						     std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
						    std::true_type,
						    std::false_type>::type;

    template <class T>
    using is_unsigned_int =
      typename std::conditional<(is_integral<T>::value &&
				 std::is_unsigned<T>::value) ||
				is_unsigned_int128<T>::value,
				std::true_type,
				std::false_type>::type;

    template <class T>
    using to_unsigned = typename std::conditional<
      is_signed_int128<T>::value,
      make_unsigned_int128<T>,
      typename std::conditional<std::is_signed<T>::value,
				std::make_unsigned<T>,
				std::common_type<T>>::type>::type;

#else

    template <class T> using is_integral = typename std::is_integral<T>;

    template <class T>
    using is_signed_int =
      typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
				std::true_type,
				std::false_type>::type;

    template <class T>
    using is_unsigned_int =
      typename std::conditional<is_integral<T>::value &&
				std::is_unsigned<T>::value,
				std::true_type,
				std::false_type>::type;

    template <class T>
    using to_unsigned = typename std::conditional<is_signed_int<T>::value,
						  std::make_unsigned<T>,
						  std::common_type<T>>::type;

#endif

    template <class T>
    using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

    template <class T>
    using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

    template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

  }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

  namespace internal {

    struct modint_base {};
    struct static_modint_base : modint_base {};

    template <class T> using is_modint = std::is_base_of<modint_base, T>;
    template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

  }  // namespace internal

  template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
  struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
      mint x;
      x._v = v;
      return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
      long long x = (long long)(v % (long long)(umod()));
      if (x < 0) x += umod();
      _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
      _v = (unsigned int)(v % umod());
    }
    static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }

    unsigned int val() const { return _v; }

    mint& operator++() {
      _v++;
      if (_v == umod()) _v = 0;
      return *this;
    }
    mint& operator--() {
      if (_v == 0) _v = umod();
      _v--;
      return *this;
    }
    mint operator++(int) {
      mint result = *this;
      ++*this;
      return result;
    }
    mint operator--(int) {
      mint result = *this;
      --*this;
      return result;
    }

    mint& operator+=(const mint& rhs) {
      _v += rhs._v;
      if (_v >= umod()) _v -= umod();
      return *this;
    }
    mint& operator-=(const mint& rhs) {
      _v -= rhs._v;
      if (_v >= umod()) _v += umod();
      return *this;
    }
    mint& operator*=(const mint& rhs) {
      unsigned long long z = _v;
      z *= rhs._v;
      _v = (unsigned int)(z % umod());
      return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
      assert(0 <= n);
      mint x = *this, r = 1;
      while (n) {
	if (n & 1) r *= x;
	x *= x;
	n >>= 1;
      }
      return r;
    }
    mint inv() const {
      if (prime) {
	assert(_v);
	return pow(umod() - 2);
      } else {
	auto eg = internal::inv_gcd(_v, m);
	assert(eg.first == 1);
	return eg.second;
      }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
      return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
      return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
      return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
      return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
      return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
      return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
  };

  template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
      assert(1 <= m);
      bt = internal::barrett(m);
    }
    static mint raw(int v) {
      mint x;
      x._v = v;
      return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
      long long x = (long long)(v % (long long)(mod()));
      if (x < 0) x += mod();
      _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
      _v = (unsigned int)(v % mod());
    }
    dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }

    unsigned int val() const { return _v; }

    mint& operator++() {
      _v++;
      if (_v == umod()) _v = 0;
      return *this;
    }
    mint& operator--() {
      if (_v == 0) _v = umod();
      _v--;
      return *this;
    }
    mint operator++(int) {
      mint result = *this;
      ++*this;
      return result;
    }
    mint operator--(int) {
      mint result = *this;
      --*this;
      return result;
    }

    mint& operator+=(const mint& rhs) {
      _v += rhs._v;
      if (_v >= umod()) _v -= umod();
      return *this;
    }
    mint& operator-=(const mint& rhs) {
      _v += mod() - rhs._v;
      if (_v >= umod()) _v -= umod();
      return *this;
    }
    mint& operator*=(const mint& rhs) {
      _v = bt.mul(_v, rhs._v);
      return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
      assert(0 <= n);
      mint x = *this, r = 1;
      while (n) {
	if (n & 1) r *= x;
	x *= x;
	n >>= 1;
      }
      return r;
    }
    mint inv() const {
      auto eg = internal::inv_gcd(_v, mod());
      assert(eg.first == 1);
      return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
      return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
      return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
      return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
      return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
      return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
      return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
  };
  template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;

  using modint998244353 = static_modint<998244353>;
  using modint1000000007 = static_modint<1000000007>;
  using modint = dynamic_modint<-1>;

  namespace internal {

    template <class T>
    using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

    template <class T>
    using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

    template <class> struct is_dynamic_modint : public std::false_type {};
    template <int id>
    struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

    template <class T>
    using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

  }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

  namespace internal {

    template <class mint, internal::is_static_modint_t<mint>* = nullptr>
    void butterfly(std::vector<mint>& a) {
      static constexpr int g = internal::primitive_root<mint::mod()>;
      int n = int(a.size());
      int h = internal::ceil_pow2(n);

      static bool first = true;
      static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
      if (first) {
        first = false;
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = bsf(mint::mod() - 1);
        mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
	  // e^(2^i) == 1
	  es[i - 2] = e;
	  ies[i - 2] = ie;
	  e *= e;
	  ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i < cnt2 - 2; i++) {
	  sum_e[i] = es[i] * now;
	  now *= ies[i];
        }
      }
      for (int ph = 1; ph <= h; ph++) {
        int w = 1 << (ph - 1), p = 1 << (h - ph);
        mint now = 1;
        for (int s = 0; s < w; s++) {
	  int offset = s << (h - ph + 1);
	  for (int i = 0; i < p; i++) {
	    auto l = a[i + offset];
	    auto r = a[i + offset + p] * now;
	    a[i + offset] = l + r;
	    a[i + offset + p] = l - r;
	  }
	  now *= sum_e[bsf(~(unsigned int)(s))];
        }
      }
    }

    template <class mint, internal::is_static_modint_t<mint>* = nullptr>
    void butterfly_inv(std::vector<mint>& a) {
      static constexpr int g = internal::primitive_root<mint::mod()>;
      int n = int(a.size());
      int h = internal::ceil_pow2(n);

      static bool first = true;
      static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
      if (first) {
        first = false;
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = bsf(mint::mod() - 1);
        mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
	  // e^(2^i) == 1
	  es[i - 2] = e;
	  ies[i - 2] = ie;
	  e *= e;
	  ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i < cnt2 - 2; i++) {
	  sum_ie[i] = ies[i] * now;
	  now *= es[i];
        }
      }

      for (int ph = h; ph >= 1; ph--) {
        int w = 1 << (ph - 1), p = 1 << (h - ph);
        mint inow = 1;
        for (int s = 0; s < w; s++) {
	  int offset = s << (h - ph + 1);
	  for (int i = 0; i < p; i++) {
	    auto l = a[i + offset];
	    auto r = a[i + offset + p];
	    a[i + offset] = l + r;
	    a[i + offset + p] =
	      (unsigned long long)(mint::mod() + l.val() - r.val()) *
	      inow.val();
	  }
	  inow *= sum_ie[bsf(~(unsigned int)(s))];
        }
      }
    }

  }  // namespace internal

  template <class mint, internal::is_static_modint_t<mint>* = nullptr>
  std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) {
      if (n < m) {
	std::swap(n, m);
	std::swap(a, b);
      }
      std::vector<mint> ans(n + m - 1);
      for (int i = 0; i < n; i++) {
	for (int j = 0; j < m; j++) {
	  ans[i + j] += a[i] * b[j];
	}
      }
      return ans;
    }
    int z = 1 << internal::ceil_pow2(n + m - 1);
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) {
      a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
    return a;
  }

  template <unsigned int mod = 998244353,
	    class T,
	    std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
  std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    using mint = static_modint<mod>;
    std::vector<mint> a2(n), b2(m);
    for (int i = 0; i < n; i++) {
      a2[i] = mint(a[i]);
    }
    for (int i = 0; i < m; i++) {
      b2[i] = mint(b[i]);
    }
    auto c2 = convolution(move(a2), move(b2));
    std::vector<T> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
      c[i] = c2[i].val();
    }
    return c;
  }

  std::vector<long long> convolution_ll(const std::vector<long long>& a,
					const std::vector<long long>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    static constexpr unsigned long long MOD1 = 754974721;  // 2^24
    static constexpr unsigned long long MOD2 = 167772161;  // 2^25
    static constexpr unsigned long long MOD3 = 469762049;  // 2^26
    static constexpr unsigned long long M2M3 = MOD2 * MOD3;
    static constexpr unsigned long long M1M3 = MOD1 * MOD3;
    static constexpr unsigned long long M1M2 = MOD1 * MOD2;
    static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

    static constexpr unsigned long long i1 =
      internal::inv_gcd(MOD2 * MOD3, MOD1).second;
    static constexpr unsigned long long i2 =
      internal::inv_gcd(MOD1 * MOD3, MOD2).second;
    static constexpr unsigned long long i3 =
      internal::inv_gcd(MOD1 * MOD2, MOD3).second;

    auto c1 = convolution<MOD1>(a, b);
    auto c2 = convolution<MOD2>(a, b);
    auto c3 = convolution<MOD3>(a, b);

    std::vector<long long> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
      unsigned long long x = 0;
      x += (c1[i] * i1) % MOD1 * M2M3;
      x += (c2[i] * i2) % MOD2 * M1M3;
      x += (c3[i] * i3) % MOD3 * M1M2;
      // B = 2^63, -B <= x, r(real value) < B
      // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
      // r = c1[i] (mod MOD1)
      // focus on MOD1
      // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
      // r = x,
      //     x - M' + (0 or 2B),
      //     x - 2M' + (0, 2B or 4B),
      //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
      // (r - x) = 0, (0)
      //           - M' + (0 or 2B), (1)
      //           -2M' + (0 or 2B or 4B), (2)
      //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
      // we checked that
      //   ((1) mod MOD1) mod 5 = 2
      //   ((2) mod MOD1) mod 5 = 3
      //   ((3) mod MOD1) mod 5 = 4
      long long diff =
	c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
      if (diff < 0) diff += MOD1;
      static constexpr unsigned long long offset[5] = {
						       0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
      x -= offset[diff % 5];
      c[i] = x;
    }

    return c;
  }

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

  // Implement (union by size) + (path compression)
  // Reference:
  // Zvi Galil and Giuseppe F. Italiano,
  // Data structures and algorithms for disjoint set union problems
  struct dsu {
  public:
    dsu() : _n(0) {}
    dsu(int n) : _n(n), parent_or_size(n, -1) {}

    int merge(int a, int b) {
      assert(0 <= a && a < _n);
      assert(0 <= b && b < _n);
      int x = leader(a), y = leader(b);
      if (x == y) return x;
      if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
      parent_or_size[x] += parent_or_size[y];
      parent_or_size[y] = x;
      return x;
    }

    bool same(int a, int b) {
      assert(0 <= a && a < _n);
      assert(0 <= b && b < _n);
      return leader(a) == leader(b);
    }

    int leader(int a) {
      assert(0 <= a && a < _n);
      if (parent_or_size[a] < 0) return a;
      return parent_or_size[a] = leader(parent_or_size[a]);
    }

    int size(int a) {
      assert(0 <= a && a < _n);
      return -parent_or_size[leader(a)];
    }

    std::vector<std::vector<int>> groups() {
      std::vector<int> leader_buf(_n), group_size(_n);
      for (int i = 0; i < _n; i++) {
	leader_buf[i] = leader(i);
	group_size[leader_buf[i]]++;
      }
      std::vector<std::vector<int>> result(_n);
      for (int i = 0; i < _n; i++) {
	result[i].reserve(group_size[i]);
      }
      for (int i = 0; i < _n; i++) {
	result[leader_buf[i]].push_back(i);
      }
      result.erase(
		   std::remove_if(result.begin(), result.end(),
				  [&](const std::vector<int>& v) { return v.empty(); }),
		   result.end());
      return result;
    }

  private:
    int _n;
    // root node: -1 * component size
    // otherwise: parent
    std::vector<int> parent_or_size;
  };

}  // namespace atcoder


#include <cassert>
#include <vector>

namespace atcoder {

  // Reference: https://en.wikipedia.org/wiki/Fenwick_tree
  template <class T> struct fenwick_tree {
    using U = internal::to_unsigned_t<T>;

  public:
    fenwick_tree() : _n(0) {}
    fenwick_tree(int n) : _n(n), data(n) {}

    void add(int p, T x) {
      assert(0 <= p && p < _n);
      p++;
      while (p <= _n) {
	data[p - 1] += U(x);
	p += p & -p;
      }
    }

    T sum(int l, int r) {
      assert(0 <= l && l <= r && r <= _n);
      return sum(r) - sum(l);
    }

  private:
    int _n;
    std::vector<U> data;

    U sum(int r) {
      U s = 0;
      while (r > 0) {
	s += data[r - 1];
	r -= r & -r;
      }
      return s;
    }
  };

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {

  template <class S,
	    S (*op)(S, S),
	    S (*e)(),
	    class F,
	    S (*mapping)(F, S),
	    F (*composition)(F, F),
	    F (*id)()>
  struct lazy_segtree {
  public:
    lazy_segtree() : lazy_segtree(0) {}
    lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
    lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
      log = internal::ceil_pow2(_n);
      size = 1 << log;
      d = std::vector<S>(2 * size, e());
      lz = std::vector<F>(size, id());
      for (int i = 0; i < _n; i++) d[size + i] = v[i];
      for (int i = size - 1; i >= 1; i--) {
	update(i);
      }
    }

    void set(int p, S x) {
      assert(0 <= p && p < _n);
      p += size;
      for (int i = log; i >= 1; i--) push(p >> i);
      d[p] = x;
      for (int i = 1; i <= log; i++) update(p >> i);
    }

    S get(int p) {
      assert(0 <= p && p < _n);
      p += size;
      for (int i = log; i >= 1; i--) push(p >> i);
      return d[p];
    }

    S prod(int l, int r) {
      assert(0 <= l && l <= r && r <= _n);
      if (l == r) return e();

      l += size;
      r += size;

      for (int i = log; i >= 1; i--) {
	if (((l >> i) << i) != l) push(l >> i);
	if (((r >> i) << i) != r) push(r >> i);
      }

      S sml = e(), smr = e();
      while (l < r) {
	if (l & 1) sml = op(sml, d[l++]);
	if (r & 1) smr = op(d[--r], smr);
	l >>= 1;
	r >>= 1;
      }

      return op(sml, smr);
    }

    S all_prod() { return d[1]; }

    void apply(int p, F f) {
      assert(0 <= p && p < _n);
      p += size;
      for (int i = log; i >= 1; i--) push(p >> i);
      d[p] = mapping(f, d[p]);
      for (int i = 1; i <= log; i++) update(p >> i);
    }
    void apply(int l, int r, F f) {
      assert(0 <= l && l <= r && r <= _n);
      if (l == r) return;

      l += size;
      r += size;

      for (int i = log; i >= 1; i--) {
	if (((l >> i) << i) != l) push(l >> i);
	if (((r >> i) << i) != r) push((r - 1) >> i);
      }

      {
	int l2 = l, r2 = r;
	while (l < r) {
	  if (l & 1) all_apply(l++, f);
	  if (r & 1) all_apply(--r, f);
	  l >>= 1;
	  r >>= 1;
	}
	l = l2;
	r = r2;
      }

      for (int i = 1; i <= log; i++) {
	if (((l >> i) << i) != l) update(l >> i);
	if (((r >> i) << i) != r) update((r - 1) >> i);
      }
    }

    template <bool (*g)(S)> int max_right(int l) {
      return max_right(l, [](S x) { return g(x); });
    }
    template <class G> int max_right(int l, G g) {
      assert(0 <= l && l <= _n);
      assert(g(e()));
      if (l == _n) return _n;
      l += size;
      for (int i = log; i >= 1; i--) push(l >> i);
      S sm = e();
      do {
	while (l % 2 == 0) l >>= 1;
	if (!g(op(sm, d[l]))) {
	  while (l < size) {
	    push(l);
	    l = (2 * l);
	    if (g(op(sm, d[l]))) {
	      sm = op(sm, d[l]);
	      l++;
	    }
	  }
	  return l - size;
	}
	sm = op(sm, d[l]);
	l++;
      } while ((l & -l) != l);
      return _n;
    }

    template <bool (*g)(S)> int min_left(int r) {
      return min_left(r, [](S x) { return g(x); });
    }
    template <class G> int min_left(int r, G g) {
      assert(0 <= r && r <= _n);
      assert(g(e()));
      if (r == 0) return 0;
      r += size;
      for (int i = log; i >= 1; i--) push((r - 1) >> i);
      S sm = e();
      do {
	r--;
	while (r > 1 && (r % 2)) r >>= 1;
	if (!g(op(d[r], sm))) {
	  while (r < size) {
	    push(r);
	    r = (2 * r + 1);
	    if (g(op(d[r], sm))) {
	      sm = op(d[r], sm);
	      r--;
	    }
	  }
	  return r + 1 - size;
	}
	sm = op(d[r], sm);
      } while ((r & -r) != r);
      return 0;
    }

  private:
    int _n, size, log;
    std::vector<S> d;
    std::vector<F> lz;

    void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
    void all_apply(int k, F f) {
      d[k] = mapping(f, d[k]);
      if (k < size) lz[k] = composition(f, lz[k]);
    }
    void push(int k) {
      all_apply(2 * k, lz[k]);
      all_apply(2 * k + 1, lz[k]);
      lz[k] = id();
    }
  };

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>

namespace atcoder {

  long long pow_mod(long long x, long long n, int m) {
    assert(0 <= n && 1 <= m);
    if (m == 1) return 0;
    internal::barrett bt((unsigned int)(m));
    unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
    while (n) {
      if (n & 1) r = bt.mul(r, y);
      y = bt.mul(y, y);
      n >>= 1;
    }
    return r;
  }

  long long inv_mod(long long x, long long m) {
    assert(1 <= m);
    auto z = internal::inv_gcd(x, m);
    assert(z.first == 1);
    return z.second;
  }

  // (rem, mod)
  std::pair<long long, long long> crt(const std::vector<long long>& r,
				      const std::vector<long long>& m) {
    assert(r.size() == m.size());
    int n = int(r.size());
    // Contracts: 0 <= r0 < m0
    long long r0 = 0, m0 = 1;
    for (int i = 0; i < n; i++) {
      assert(1 <= m[i]);
      long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
      if (m0 < m1) {
	std::swap(r0, r1);
	std::swap(m0, m1);
      }
      if (m0 % m1 == 0) {
	if (r0 % m1 != r1) return {0, 0};
	continue;
      }
      // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

      // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
      // r2 % m0 = r0
      // r2 % m1 = r1
      // -> (r0 + x*m0) % m1 = r1
      // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
      // -> x = (r1 - r0) / g * inv(u0) (mod u1)

      // im = inv(u0) (mod u1) (0 <= im < u1)
      long long g, im;
      std::tie(g, im) = internal::inv_gcd(m0, m1);

      long long u1 = (m1 / g);
      // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
      if ((r1 - r0) % g) return {0, 0};

      // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
      long long x = (r1 - r0) / g % u1 * im % u1;

      // |r0| + |m0 * x|
      // < m0 + m0 * (u1 - 1)
      // = m0 + m0 * m1 / g - m0
      // = lcm(m0, m1)
      r0 += x * m0;
      m0 *= u1;  // -> lcm(m0, m1)
      if (r0 < 0) r0 += m0;
    }
    return {r0, m0};
  }

  long long floor_sum(long long n, long long m, long long a, long long b) {
    long long ans = 0;
    if (a >= m) {
      ans += (n - 1) * n * (a / m) / 2;
      a %= m;
    }
    if (b >= m) {
      ans += n * (b / m);
      b %= m;
    }

    long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
    if (y_max == 0) return ans;
    ans += (n - (x_max + a - 1) / a) * y_max;
    ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
    return ans;
  }

}  // namespace atcoder


#include <algorithm>

#include <vector>

namespace atcoder {

  namespace internal {

    template <class T> struct simple_queue {
      std::vector<T> payload;
      int pos = 0;
      void reserve(int n) { payload.reserve(n); }
      int size() const { return int(payload.size()) - pos; }
      bool empty() const { return pos == int(payload.size()); }
      void push(const T& t) { payload.push_back(t); }
      T& front() { return payload[pos]; }
      void clear() {
        payload.clear();
        pos = 0;
      }
      void pop() { pos++; }
    };

  }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

  template <class Cap> struct mf_graph {
  public:
    mf_graph() : _n(0) {}
    mf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap) {
      assert(0 <= from && from < _n);
      assert(0 <= to && to < _n);
      assert(0 <= cap);
      int m = int(pos.size());
      pos.push_back({from, int(g[from].size())});
      g[from].push_back(_edge{to, int(g[to].size()), cap});
      g[to].push_back(_edge{from, int(g[from].size()) - 1, 0});
      return m;
    }

    struct edge {
      int from, to;
      Cap cap, flow;
    };

    edge get_edge(int i) {
      int m = int(pos.size());
      assert(0 <= i && i < m);
      auto _e = g[pos[i].first][pos[i].second];
      auto _re = g[_e.to][_e.rev];
      return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    std::vector<edge> edges() {
      int m = int(pos.size());
      std::vector<edge> result;
      for (int i = 0; i < m; i++) {
	result.push_back(get_edge(i));
      }
      return result;
    }
    void change_edge(int i, Cap new_cap, Cap new_flow) {
      int m = int(pos.size());
      assert(0 <= i && i < m);
      assert(0 <= new_flow && new_flow <= new_cap);
      auto& _e = g[pos[i].first][pos[i].second];
      auto& _re = g[_e.to][_e.rev];
      _e.cap = new_cap - new_flow;
      _re.cap = new_flow;
    }

    Cap flow(int s, int t) {
      return flow(s, t, std::numeric_limits<Cap>::max());
    }
    Cap flow(int s, int t, Cap flow_limit) {
      assert(0 <= s && s < _n);
      assert(0 <= t && t < _n);

      std::vector<int> level(_n), iter(_n);
      internal::simple_queue<int> que;

      auto bfs = [&]() {
		   std::fill(level.begin(), level.end(), -1);
		   level[s] = 0;
		   que.clear();
		   que.push(s);
		   while (!que.empty()) {
		     int v = que.front();
		     que.pop();
		     for (auto e : g[v]) {
		       if (e.cap == 0 || level[e.to] >= 0) continue;
		       level[e.to] = level[v] + 1;
		       if (e.to == t) return;
		       que.push(e.to);
		     }
		   }
		 };
      auto dfs = [&](auto self, int v, Cap up) {
		   if (v == s) return up;
		   Cap res = 0;
		   int level_v = level[v];
		   for (int& i = iter[v]; i < int(g[v].size()); i++) {
		     _edge& e = g[v][i];
		     if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
		     Cap d =
		       self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
		     if (d <= 0) continue;
		     g[v][i].cap += d;
		     g[e.to][e.rev].cap -= d;
		     res += d;
		     if (res == up) break;
		   }
		   return res;
		 };

      Cap flow = 0;
      while (flow < flow_limit) {
	bfs();
	if (level[t] == -1) break;
	std::fill(iter.begin(), iter.end(), 0);
	while (flow < flow_limit) {
	  Cap f = dfs(dfs, t, flow_limit - flow);
	  if (!f) break;
	  flow += f;
	}
      }
      return flow;
    }

    std::vector<bool> min_cut(int s) {
      std::vector<bool> visited(_n);
      internal::simple_queue<int> que;
      que.push(s);
      while (!que.empty()) {
	int p = que.front();
	que.pop();
	visited[p] = true;
	for (auto e : g[p]) {
	  if (e.cap && !visited[e.to]) {
	    visited[e.to] = true;
	    que.push(e.to);
	  }
	}
      }
      return visited;
    }

  private:
    int _n;
    struct _edge {
      int to, rev;
      Cap cap;
    };
    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
  };

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

  template <class Cap, class Cost> struct mcf_graph {
  public:
    mcf_graph() {}
    mcf_graph(int n) : _n(n), g(n) {}

    int add_edge(int from, int to, Cap cap, Cost cost) {
      assert(0 <= from && from < _n);
      assert(0 <= to && to < _n);
      int m = int(pos.size());
      pos.push_back({from, int(g[from].size())});
      g[from].push_back(_edge{to, int(g[to].size()), cap, cost});
      g[to].push_back(_edge{from, int(g[from].size()) - 1, 0, -cost});
      return m;
    }

    struct edge {
      int from, to;
      Cap cap, flow;
      Cost cost;
    };

    edge get_edge(int i) {
      int m = int(pos.size());
      assert(0 <= i && i < m);
      auto _e = g[pos[i].first][pos[i].second];
      auto _re = g[_e.to][_e.rev];
      return edge{
	pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
	  };
    }
    std::vector<edge> edges() {
      int m = int(pos.size());
      std::vector<edge> result(m);
      for (int i = 0; i < m; i++) {
	result[i] = get_edge(i);
      }
      return result;
    }

    std::pair<Cap, Cost> flow(int s, int t) {
      return flow(s, t, std::numeric_limits<Cap>::max());
    }
    std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
      return slope(s, t, flow_limit).back();
    }
    std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
      return slope(s, t, std::numeric_limits<Cap>::max());
    }
    std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
      assert(0 <= s && s < _n);
      assert(0 <= t && t < _n);
      assert(s != t);
      // variants (C = maxcost):
      // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0
      // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge
      std::vector<Cost> dual(_n, 0), dist(_n);
      std::vector<int> pv(_n), pe(_n);
      std::vector<bool> vis(_n);
      auto dual_ref = [&]() {
			std::fill(dist.begin(), dist.end(),
				  std::numeric_limits<Cost>::max());
			std::fill(pv.begin(), pv.end(), -1);
			std::fill(pe.begin(), pe.end(), -1);
			std::fill(vis.begin(), vis.end(), false);
			struct Q {
			  Cost key;
			  int to;
			  bool operator<(Q r) const { return key > r.key; }
			};
			std::priority_queue<Q> que;
			dist[s] = 0;
			que.push(Q{0, s});
			while (!que.empty()) {
			  int v = que.top().to;
			  que.pop();
			  if (vis[v]) continue;
			  vis[v] = true;
			  if (v == t) break;
			  // dist[v] = shortest(s, v) + dual[s] - dual[v]
			  // dist[v] >= 0 (all reduced cost are positive)
			  // dist[v] <= (n-1)C
			  for (int i = 0; i < int(g[v].size()); i++) {
			    auto e = g[v][i];
			    if (vis[e.to] || !e.cap) continue;
			    // |-dual[e.to] + dual[v]| <= (n-1)C
			    // cost <= C - -(n-1)C + 0 = nC
			    Cost cost = e.cost - dual[e.to] + dual[v];
			    if (dist[e.to] - dist[v] > cost) {
			      dist[e.to] = dist[v] + cost;
			      pv[e.to] = v;
			      pe[e.to] = i;
			      que.push(Q{dist[e.to], e.to});
			    }
			  }
			}
			if (!vis[t]) {
			  return false;
			}

			for (int v = 0; v < _n; v++) {
			  if (!vis[v]) continue;
			  // dual[v] = dual[v] - dist[t] + dist[v]
			  //         = dual[v] - (shortest(s, t) + dual[s] - dual[t]) + (shortest(s, v) + dual[s] - dual[v])
			  //         = - shortest(s, t) + dual[t] + shortest(s, v)
			  //         = shortest(s, v) - shortest(s, t) >= 0 - (n-1)C
			  dual[v] -= dist[t] - dist[v];
			}
			return true;
		      };
      Cap flow = 0;
      Cost cost = 0, prev_cost = -1;
      std::vector<std::pair<Cap, Cost>> result;
      result.push_back({flow, cost});
      while (flow < flow_limit) {
	if (!dual_ref()) break;
	Cap c = flow_limit - flow;
	for (int v = t; v != s; v = pv[v]) {
	  c = std::min(c, g[pv[v]][pe[v]].cap);
	}
	for (int v = t; v != s; v = pv[v]) {
	  auto& e = g[pv[v]][pe[v]];
	  e.cap -= c;
	  g[v][e.rev].cap += c;
	}
	Cost d = -dual[s];
	flow += c;
	cost += c * d;
	if (prev_cost == d) {
	  result.pop_back();
	}
	result.push_back({flow, cost});
	prev_cost = cost;
      }
      return result;
    }

  private:
    int _n;

    struct _edge {
      int to, rev;
      Cap cap;
      Cost cost;
    };

    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
  };

}  // namespace atcoder


#include <algorithm>

#include <algorithm>
#include <utility>
#include <vector>

namespace atcoder {
  namespace internal {

    template <class E> struct csr {
      std::vector<int> start;
      std::vector<E> elist;
      csr(int n, const std::vector<std::pair<int, E>>& edges)
        : start(n + 1), elist(edges.size()) {
        for (auto e : edges) {
	  start[e.first + 1]++;
        }
        for (int i = 1; i <= n; i++) {
	  start[i] += start[i - 1];
        }
        auto counter = start;
        for (auto e : edges) {
	  elist[counter[e.first]++] = e.second;
        }
      }
    };

    // Reference:
    // R. Tarjan,
    // Depth-First Search and Linear Graph Algorithms
    struct scc_graph {
    public:
      scc_graph(int n) : _n(n) {}

      int num_vertices() { return _n; }

      void add_edge(int from, int to) { edges.push_back({from, {to}}); }

      // @return pair of (# of scc, scc id)
      std::pair<int, std::vector<int>> scc_ids() {
        auto g = csr<edge>(_n, edges);
        int now_ord = 0, group_num = 0;
        std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
        visited.reserve(_n);
        auto dfs = [&](auto self, int v) -> void {
		     low[v] = ord[v] = now_ord++;
		     visited.push_back(v);
		     for (int i = g.start[v]; i < g.start[v + 1]; i++) {
		       auto to = g.elist[i].to;
		       if (ord[to] == -1) {
			 self(self, to);
			 low[v] = std::min(low[v], low[to]);
		       } else {
			 low[v] = std::min(low[v], ord[to]);
		       }
		     }
		     if (low[v] == ord[v]) {
		       while (true) {
			 int u = visited.back();
			 visited.pop_back();
			 ord[u] = _n;
			 ids[u] = group_num;
			 if (u == v) break;
		       }
		       group_num++;
		     }
		   };
        for (int i = 0; i < _n; i++) {
	  if (ord[i] == -1) dfs(dfs, i);
        }
        for (auto& x : ids) {
	  x = group_num - 1 - x;
        }
        return {group_num, ids};
      }

      std::vector<std::vector<int>> scc() {
        auto ids = scc_ids();
        int group_num = ids.first;
        std::vector<int> counts(group_num);
        for (auto x : ids.second) counts[x]++;
        std::vector<std::vector<int>> groups(ids.first);
        for (int i = 0; i < group_num; i++) {
	  groups[i].reserve(counts[i]);
        }
        for (int i = 0; i < _n; i++) {
	  groups[ids.second[i]].push_back(i);
        }
        return groups;
      }

    private:
      int _n;
      struct edge {
        int to;
      };
      std::vector<std::pair<int, edge>> edges;
    };

  }  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

  struct scc_graph {
  public:
    scc_graph() : internal(0) {}
    scc_graph(int n) : internal(n) {}

    void add_edge(int from, int to) {
      int n = internal.num_vertices();
      assert(0 <= from && from < n);
      assert(0 <= to && to < n);
      internal.add_edge(from, to);
    }

    std::vector<std::vector<int>> scc() { return internal.scc(); }

  private:
    internal::scc_graph internal;
  };

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <vector>

namespace atcoder {

  template <class S, S (*op)(S, S), S (*e)()> struct segtree {
  public:
    segtree() : segtree(0) {}
    segtree(int n) : segtree(std::vector<S>(n, e())) {}
    segtree(const std::vector<S>& v) : _n(int(v.size())) {
      log = internal::ceil_pow2(_n);
      size = 1 << log;
      d = std::vector<S>(2 * size, e());
      for (int i = 0; i < _n; i++) d[size + i] = v[i];
      for (int i = size - 1; i >= 1; i--) {
	update(i);
      }
    }

    void set(int p, S x) {
      assert(0 <= p && p < _n);
      p += size;
      d[p] = x;
      for (int i = 1; i <= log; i++) update(p >> i);
    }

    S get(int p) {
      assert(0 <= p && p < _n);
      return d[p + size];
    }

    S prod(int l, int r) {
      assert(0 <= l && l <= r && r <= _n);
      S sml = e(), smr = e();
      l += size;
      r += size;

      while (l < r) {
	if (l & 1) sml = op(sml, d[l++]);
	if (r & 1) smr = op(d[--r], smr);
	l >>= 1;
	r >>= 1;
      }
      return op(sml, smr);
    }

    S all_prod() { return d[1]; }

    template <bool (*f)(S)> int max_right(int l) {
      return max_right(l, [](S x) { return f(x); });
    }
    template <class F> int max_right(int l, F f) {
      assert(0 <= l && l <= _n);
      assert(f(e()));
      if (l == _n) return _n;
      l += size;
      S sm = e();
      do {
	while (l % 2 == 0) l >>= 1;
	if (!f(op(sm, d[l]))) {
	  while (l < size) {
	    l = (2 * l);
	    if (f(op(sm, d[l]))) {
	      sm = op(sm, d[l]);
	      l++;
	    }
	  }
	  return l - size;
	}
	sm = op(sm, d[l]);
	l++;
      } while ((l & -l) != l);
      return _n;
    }

    template <bool (*f)(S)> int min_left(int r) {
      return min_left(r, [](S x) { return f(x); });
    }
    template <class F> int min_left(int r, F f) {
      assert(0 <= r && r <= _n);
      assert(f(e()));
      if (r == 0) return 0;
      r += size;
      S sm = e();
      do {
	r--;
	while (r > 1 && (r % 2)) r >>= 1;
	if (!f(op(d[r], sm))) {
	  while (r < size) {
	    r = (2 * r + 1);
	    if (f(op(d[r], sm))) {
	      sm = op(d[r], sm);
	      r--;
	    }
	  }
	  return r + 1 - size;
	}
	sm = op(d[r], sm);
      } while ((r & -r) != r);
      return 0;
    }

  private:
    int _n, size, log;
    std::vector<S> d;

    void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
  };

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>

namespace atcoder {

  namespace internal {

    std::vector<int> sa_naive(const std::vector<int>& s) {
      int n = int(s.size());
      std::vector<int> sa(n);
      std::iota(sa.begin(), sa.end(), 0);
      std::sort(sa.begin(), sa.end(), [&](int l, int r) {
					if (l == r) return false;
					while (l < n && r < n) {
					  if (s[l] != s[r]) return s[l] < s[r];
					  l++;
					  r++;
					}
					return l == n;
				      });
      return sa;
    }

    std::vector<int> sa_doubling(const std::vector<int>& s) {
      int n = int(s.size());
      std::vector<int> sa(n), rnk = s, tmp(n);
      std::iota(sa.begin(), sa.end(), 0);
      for (int k = 1; k < n; k *= 2) {
        auto cmp = [&](int x, int y) {
		     if (rnk[x] != rnk[y]) return rnk[x] < rnk[y];
		     int rx = x + k < n ? rnk[x + k] : -1;
		     int ry = y + k < n ? rnk[y + k] : -1;
		     return rx < ry;
		   };
        std::sort(sa.begin(), sa.end(), cmp);
        tmp[sa[0]] = 0;
        for (int i = 1; i < n; i++) {
	  tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
        }
        std::swap(tmp, rnk);
      }
      return sa;
    }

    // SA-IS, linear-time suffix array construction
    // Reference:
    // G. Nong, S. Zhang, and W. H. Chan,
    // Two Efficient Algorithms for Linear Time Suffix Array Construction
    template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
    std::vector<int> sa_is(const std::vector<int>& s, int upper) {
      int n = int(s.size());
      if (n == 0) return {};
      if (n == 1) return {0};
      if (n == 2) {
        if (s[0] < s[1]) {
	  return {0, 1};
        } else {
	  return {1, 0};
        }
      }
      if (n < THRESHOLD_NAIVE) {
        return sa_naive(s);
      }
      if (n < THRESHOLD_DOUBLING) {
        return sa_doubling(s);
      }

      std::vector<int> sa(n);
      std::vector<bool> ls(n);
      for (int i = n - 2; i >= 0; i--) {
        ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
      }
      std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
      for (int i = 0; i < n; i++) {
        if (!ls[i]) {
	  sum_s[s[i]]++;
        } else {
	  sum_l[s[i] + 1]++;
        }
      }
      for (int i = 0; i <= upper; i++) {
        sum_s[i] += sum_l[i];
        if (i < upper) sum_l[i + 1] += sum_s[i];
      }

      auto induce = [&](const std::vector<int>& lms) {
		      std::fill(sa.begin(), sa.end(), -1);
		      std::vector<int> buf(upper + 1);
		      std::copy(sum_s.begin(), sum_s.end(), buf.begin());
		      for (auto d : lms) {
			if (d == n) continue;
			sa[buf[s[d]]++] = d;
		      }
		      std::copy(sum_l.begin(), sum_l.end(), buf.begin());
		      sa[buf[s[n - 1]]++] = n - 1;
		      for (int i = 0; i < n; i++) {
			int v = sa[i];
			if (v >= 1 && !ls[v - 1]) {
			  sa[buf[s[v - 1]]++] = v - 1;
			}
		      }
		      std::copy(sum_l.begin(), sum_l.end(), buf.begin());
		      for (int i = n - 1; i >= 0; i--) {
			int v = sa[i];
			if (v >= 1 && ls[v - 1]) {
			  sa[--buf[s[v - 1] + 1]] = v - 1;
			}
		      }
		    };

      std::vector<int> lms_map(n + 1, -1);
      int m = 0;
      for (int i = 1; i < n; i++) {
        if (!ls[i - 1] && ls[i]) {
	  lms_map[i] = m++;
        }
      }
      std::vector<int> lms;
      lms.reserve(m);
      for (int i = 1; i < n; i++) {
        if (!ls[i - 1] && ls[i]) {
	  lms.push_back(i);
        }
      }

      induce(lms);

      if (m) {
        std::vector<int> sorted_lms;
        sorted_lms.reserve(m);
        for (int v : sa) {
	  if (lms_map[v] != -1) sorted_lms.push_back(v);
        }
        std::vector<int> rec_s(m);
        int rec_upper = 0;
        rec_s[lms_map[sorted_lms[0]]] = 0;
        for (int i = 1; i < m; i++) {
	  int l = sorted_lms[i - 1], r = sorted_lms[i];
	  int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
	  int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
	  bool same = true;
	  if (end_l - l != end_r - r) {
	    same = false;
	  } else {
	    while (l < end_l) {
	      if (s[l] != s[r]) {
		break;
	      }
	      l++;
	      r++;
	    }
	    if (l == n || s[l] != s[r]) same = false;
	  }
	  if (!same) rec_upper++;
	  rec_s[lms_map[sorted_lms[i]]] = rec_upper;
        }

        auto rec_sa =
	  sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);

        for (int i = 0; i < m; i++) {
	  sorted_lms[i] = lms[rec_sa[i]];
        }
        induce(sorted_lms);
      }
      return sa;
    }

  }  // namespace internal

  std::vector<int> suffix_array(const std::vector<int>& s, int upper) {
    assert(0 <= upper);
    for (int d : s) {
      assert(0 <= d && d <= upper);
    }
    auto sa = internal::sa_is(s, upper);
    return sa;
  }

  template <class T> std::vector<int> suffix_array(const std::vector<T>& s) {
    int n = int(s.size());
    std::vector<int> idx(n);
    iota(idx.begin(), idx.end(), 0);
    sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
    std::vector<int> s2(n);
    int now = 0;
    for (int i = 0; i < n; i++) {
      if (i && s[idx[i - 1]] != s[idx[i]]) now++;
      s2[idx[i]] = now;
    }
    return internal::sa_is(s2, now);
  }

  std::vector<int> suffix_array(const std::string& s) {
    int n = int(s.size());
    std::vector<int> s2(n);
    for (int i = 0; i < n; i++) {
      s2[i] = s[i];
    }
    return internal::sa_is(s2, 255);
  }

  // Reference:
  // T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,
  // Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its
  // Applications
  template <class T>
  std::vector<int> lcp_array(const std::vector<T>& s,
			     const std::vector<int>& sa) {
    int n = int(s.size());
    assert(n >= 1);
    std::vector<int> rnk(n);
    for (int i = 0; i < n; i++) {
      rnk[sa[i]] = i;
    }
    std::vector<int> lcp(n - 1);
    int h = 0;
    for (int i = 0; i < n; i++) {
      if (h > 0) h--;
      if (rnk[i] == 0) continue;
      int j = sa[rnk[i] - 1];
      for (; j + h < n && i + h < n; h++) {
	if (s[j + h] != s[i + h]) break;
      }
      lcp[rnk[i] - 1] = h;
    }
    return lcp;
  }

  std::vector<int> lcp_array(const std::string& s, const std::vector<int>& sa) {
    int n = int(s.size());
    std::vector<int> s2(n);
    for (int i = 0; i < n; i++) {
      s2[i] = s[i];
    }
    return lcp_array(s2, sa);
  }

  // Reference:
  // D. Gusfield,
  // Algorithms on Strings, Trees, and Sequences: Computer Science and
  // Computational Biology
  template <class T> std::vector<int> z_algorithm(const std::vector<T>& s) {
    int n = int(s.size());
    if (n == 0) return {};
    std::vector<int> z(n);
    z[0] = 0;
    for (int i = 1, j = 0; i < n; i++) {
      int& k = z[i];
      k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
      while (i + k < n && s[k] == s[i + k]) k++;
      if (j + z[j] < i + z[i]) j = i;
    }
    z[0] = n;
    return z;
  }

  std::vector<int> z_algorithm(const std::string& s) {
    int n = int(s.size());
    std::vector<int> s2(n);
    for (int i = 0; i < n; i++) {
      s2[i] = s[i];
    }
    return z_algorithm(s2);
  }

}  // namespace atcoder


#include <cassert>
#include <vector>

namespace atcoder {

  // Reference:
  // B. Aspvall, M. Plass, and R. Tarjan,
  // A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
  // Formulas
  struct two_sat {
  public:
    two_sat() : _n(0), scc(0) {}
    two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}

    void add_clause(int i, bool f, int j, bool g) {
      assert(0 <= i && i < _n);
      assert(0 <= j && j < _n);
      scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
      scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
    }
    bool satisfiable() {
      auto id = scc.scc_ids().second;
      for (int i = 0; i < _n; i++) {
	if (id[2 * i] == id[2 * i + 1]) return false;
	_answer[i] = id[2 * i] < id[2 * i + 1];
      }
      return true;
    }
    std::vector<bool> answer() { return _answer; }

  private:
    int _n;
    std::vector<bool> _answer;
    internal::scc_graph scc;
  };

}  // namespace atcoder


using namespace std;
using namespace atcoder;

#define debug(n) cerr << #n << ':' << n << endl;
#define dline    cerr << __LINE__ << endl;

using ll = long long;
template<class T, class U> using P = pair<T,U>;
template<class T> using Heap = priority_queue<T>;
template<class T> using heaP = priority_queue<T,vector<T>,greater<T>>;
template<class T,class U> using umap = unordered_map<T,U>;
template<class T> using uset = unordered_set<T>;

template<class T>
bool ChangeMax(T&a,const T&b){
  if(a >= b) return false;
  a = b;    return true;
}

template<class T>
bool ChangeMin(T&a,const T&b){
  if(a <= b) return false;
  a = b;    return true;  
}

template<class T, size_t N, class U>
void Fill(T (&a)[N], const U&v){
  fill((U*)a,(U*)(a+N),v);
}

template<class T>
istream& operator >> (istream&is, vector<T>&v){
  for(auto&e:v)is >> e;
  return is;
}

int op(int a,int b){return max(a,b);}
int e(){return 0;}

int main(){
  int n,k; cin >> n >> k;
  segtree<int,op,e> sgt(300001);
  vector<int> dp(n);
  int ans = 0;
  for(int i = 0; i < n; ++i){
    int x; cin >> x;
    int y = sgt.prod(max(x-k,0), min(x+k+1,300001));
    dp[i] = y + 1;
    ans = max(ans,dp[i]);
    sgt.set(x, dp[i]);
  }
  cout << ans << endl;
  return 0;
}


Measured memory_bytes: 8592