#line 2 "/home/shogo314/cpp_include/ou-library/unionfind.hpp"

/**
 * @file unionfind.hpp
 * @brief UnionFind
 */

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

/**
 * @brief 無向グラフに対して「辺の追加」、「2頂点が連結かの判定」をする
 */
struct UnionFind {
   private:
    int _n;
    // 負ならサイズ、非負なら親
    std::vector<int> parent_or_size;

   public:
    UnionFind() : _n(0) {}
    explicit UnionFind(int n) : _n(n), parent_or_size(n, -1) {}

    /**
     * @brief 辺(a,b)を足す
     * @return 連結したものの代表元
     */
    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;
    }
    /**
     * @brief 頂点a,bが連結かどうか
     */
    bool same(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        return leader(a) == leader(b);
    }
    /**
     * @brief 頂点aの属する連結成分の代表元
     */
    int leader(int a) {
        assert(0 <= a && a < _n);
        if (parent_or_size[a] < 0) return a;
        int x = a;
        while (parent_or_size[x] >= 0) {
            x = parent_or_size[x];
        }
        while (parent_or_size[a] >= 0) {
            int t = parent_or_size[a];
            parent_or_size[a] = x;
            a = t;
        }
        return x;
    }
    /**
     * @brief 頂点aの属する連結成分のサイズ
     */
    int size(int a) {
        assert(0 <= a && a < _n);
        return -parent_or_size[leader(a)];
    }
    /**
     * @brief グラフを連結成分に分け、その情報を返す
     * @return 「一つの連結成分の頂点番号のリスト」のリスト
     */
    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;
    }
};
#line 2 "/home/shogo314/cpp_include/sh-library/base/all"
#include <bits/stdc++.h>
#line 5 "/home/shogo314/cpp_include/sh-library/base/container_func.hpp"
#include <initializer_list>
#line 3 "/home/shogo314/cpp_include/sh-library/base/traits.hpp"
#include <type_traits>

#define HAS_METHOD(func_name)                                                              \
    namespace detail {                                                                     \
    template <class T, class = void>                                                       \
    struct has_##func_name##_impl : std::false_type {};                                    \
    template <class T>                                                                     \
    struct has_##func_name##_impl<T, std::void_t<decltype(std::declval<T>().func_name())>> \
        : std::true_type {};                                                               \
    }                                                                                      \
    template <class T>                                                                     \
    struct has_##func_name : detail::has_##func_name##_impl<T>::type {};                   \
    template <class T>                                                                     \
    inline constexpr bool has_##func_name##_v = has_##func_name<T>::value;

#define HAS_METHOD_ARG(func_name)                                                                              \
    namespace detail {                                                                                         \
    template <class T, typename U, class = void>                                                               \
    struct has_##func_name##_impl : std::false_type {};                                                        \
    template <class T, typename U>                                                                             \
    struct has_##func_name##_impl<T, U, std::void_t<decltype(std::declval<T>().func_name(std::declval<U>()))>> \
        : std::true_type {};                                                                                   \
    }                                                                                                          \
    template <class T, typename U>                                                                             \
    struct has_##func_name : detail::has_##func_name##_impl<T, U>::type {};                                    \
    template <class T, typename U>                                                                             \
    inline constexpr bool has_##func_name##_v = has_##func_name<T, U>::value;

HAS_METHOD(repr)
HAS_METHOD(type_str)
HAS_METHOD(initializer_str)
HAS_METHOD(max)
HAS_METHOD(min)
HAS_METHOD(reversed)
HAS_METHOD(sorted)
HAS_METHOD(sum)
HAS_METHOD(product)
HAS_METHOD_ARG(count)
HAS_METHOD_ARG(find)
HAS_METHOD_ARG(lower_bound)
HAS_METHOD_ARG(upper_bound)

#define ENABLE_IF_T_IMPL(expr) std::enable_if_t<expr, std::nullptr_t> = nullptr
#define ENABLE_IF_T(...) ENABLE_IF_T_IMPL((__VA_ARGS__))

template <class C>
using mem_value_type = typename C::value_type;
template <class C>
using mem_difference_type = typename C::difference_type;
#line 9 "/home/shogo314/cpp_include/sh-library/base/container_func.hpp"

#define METHOD_EXPAND(func_name)                                             \
    template <typename T, ENABLE_IF_T(has_##func_name##_v<T>)>               \
    inline constexpr auto func_name(const T &t) -> decltype(t.func_name()) { \
        return t.func_name();                                                \
    }

#define METHOD_AND_FUNC_ARG_EXPAND(func_name)                                     \
    template <typename T, typename U, ENABLE_IF_T(has_##func_name##_v<T, U>)>     \
    inline constexpr auto func_name(const T &t, const U &u)                       \
        -> decltype(t.func_name(u)) {                                             \
        return t.func_name(u);                                                    \
    }                                                                             \
    template <typename T, typename U, ENABLE_IF_T(not has_##func_name##_v<T, U>)> \
    inline constexpr auto func_name(const T &t, const U &u)                       \
        -> decltype(func_name(t.begin(), t.end(), u)) {                           \
        using namespace std;                                                      \
        return func_name(t.begin(), t.end(), u);                                  \
    }

METHOD_EXPAND(reversed)
template <class C, ENABLE_IF_T(not has_reversed_v<C>)>
inline constexpr C reversed(C t) {
    std::reverse(t.begin(), t.end());
    return t;
}

METHOD_EXPAND(sorted)
template <class C, ENABLE_IF_T(not has_sorted_v<C>)>
inline constexpr C sorted(C t) {
    std::sort(t.begin(), t.end());
    return t;
}
template <class C, class F, ENABLE_IF_T(not has_sorted_v<C> and std::is_invocable_r_v<bool, F, mem_value_type<C>, mem_value_type<C>>)>
inline constexpr C sorted(C t, F f) {
    std::sort(t.begin(), t.end(), f);
    return t;
}

template <class C>
inline constexpr void sort(C &t) {
    std::sort(t.begin(), t.end());
}
template <class C, class F, ENABLE_IF_T(std::is_invocable_r_v<bool, F, mem_value_type<C>, mem_value_type<C>>)>
inline constexpr void sort(C &t, F f) {
    std::sort(t.begin(), t.end(), f);
}
template <class C, class F, ENABLE_IF_T(std::is_invocable_v<F, mem_value_type<C>>)>
inline constexpr void sort_by_key(C &t, F f) {
    std::sort(t.begin(), t.end(), [&](const mem_value_type<C> &left, const mem_value_type<C> &right) {
        return f(left) < f(right);
    });
}

template <class C>
inline constexpr void reverse(C &t) {
    std::reverse(t.begin(), t.end());
}

METHOD_EXPAND(max)
template <class C, ENABLE_IF_T(not has_max_v<C>)>
inline constexpr mem_value_type<C> max(const C &v) {
    assert(v.begin() != v.end());
    return *std::max_element(v.begin(), v.end());
}
template <typename T>
inline constexpr T max(const std::initializer_list<T> &v) {
    return std::max(v);
}

METHOD_EXPAND(min)
template <class C, ENABLE_IF_T(not has_max_v<C>)>
inline constexpr mem_value_type<C> min(const C &v) {
    assert(v.begin() != v.end());
    return *std::min_element(v.begin(), v.end());
}
template <typename T>
inline constexpr T min(const std::initializer_list<T> &v) {
    return std::min(v);
}

METHOD_EXPAND(sum)
template <class C, ENABLE_IF_T(not has_sum_v<C>)>
inline constexpr mem_value_type<C> sum(const C &v) {
    return std::accumulate(v.begin(), v.end(), mem_value_type<C>{});
}
template <typename T>
inline constexpr T sum(const std::initializer_list<T> &v) {
    return std::accumulate(v.begin(), v.end(), T{});
}

METHOD_EXPAND(product)
template <class C, ENABLE_IF_T(not has_product_v<C>)>
inline constexpr mem_value_type<C> product(const C &v) {
    return std::accumulate(v.begin(), v.end(), mem_value_type<C>{1}, std::multiplies<mem_value_type<C>>());
}
template <typename T>
inline constexpr T product(const std::initializer_list<T> &v) {
    return std::accumulate(v.begin(), v.end(), T{1}, std::multiplies<T>());
}

template <class C>
inline constexpr mem_value_type<C> product_xor(const C &v) {
    return std::accumulate(v.begin(), v.end(), mem_value_type<C>{0}, std::bit_xor<mem_value_type<C>>());
}
template <typename T>
inline constexpr T product_xor(const std::initializer_list<T> &v) {
    return std::accumulate(v.begin(), v.end(), T{0}, std::bit_xor<T>());
}

METHOD_AND_FUNC_ARG_EXPAND(count)
METHOD_AND_FUNC_ARG_EXPAND(find)
METHOD_AND_FUNC_ARG_EXPAND(lower_bound)
METHOD_AND_FUNC_ARG_EXPAND(upper_bound)

template <class C>
inline constexpr mem_value_type<C> gcd(const C &v) {
    mem_value_type<C> init(0);
    for (const auto &e : v) init = std::gcd(init, e);
    return init;
}

template <class C>
inline constexpr mem_value_type<C> average(const C &v) {
    assert(v.size());
    return sum(v) / v.size();
}

template <class C>
inline constexpr mem_value_type<C> median(const C &v) {
    assert(not v.empty());
    std::vector<size_t> u(v.size());
    std::iota(u.begin(), u.end(), 0);
    std::sort(u.begin(), u.end(), [&](size_t a, size_t b) {
        return v[a] < v[b];
    });
    if (v.size() & 1) {
        return v[u[v.size() / 2]];
    }
    // C++20
    // return std::midpoint(v[u[v.size() / 2]], v[u[v.size() / 2 - 1]]);
    return (v[u[v.size() / 2]] + v[u[v.size() / 2 - 1]]) / 2;
}

template <class C, typename U>
inline constexpr size_t index(const C &v, const U &x) {
    return std::distance(v.begin(), std::find(v.begin(), v.end(), x));
}

template <class C, ENABLE_IF_T(std::is_integral_v<mem_value_type<C>>)>
inline constexpr mem_value_type<C> mex(const C &v) {
    std::vector<bool> b(v.size() + 1);
    for (const auto &a : v) {
        if (0 <= a and a < b.size()) {
            b[a] = true;
        }
    }
    mem_value_type<C> ret;
    for (size_t i = 0; i < b.size(); i++) {
        if (not b[i]) {
            ret = i;
            break;
        }
    }
    return ret;
}

template <class C>
inline constexpr mem_difference_type<C> bisect_left(const C &v, const mem_value_type<C> &x) {
    return std::distance(v.begin(), lower_bound(v, x));
}
template <class C>
inline constexpr mem_difference_type<C> bisect_right(const C &v, const mem_value_type<C> &x) {
    return std::distance(v.begin(), upper_bound(v, x));
}
#line 6 "/home/shogo314/cpp_include/sh-library/base/functions.hpp"

template <typename T1, typename T2>
inline constexpr bool chmin(T1 &a, T2 b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}

template <typename T1, typename T2>
inline constexpr bool chmax(T1 &a, T2 b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

template <class C, typename T>
inline constexpr bool contains(const C &c, const T &t) {
    return c.find(t) != c.end();
}

inline constexpr long long max(const long long &t1, const long long &t2) {
    return std::max<long long>(t1, t2);
}

inline constexpr long long min(const long long &t1, const long long &t2) {
    return std::min<long long>(t1, t2);
}

using std::abs;
using std::gcd;
using std::lcm;
using std::size;

template <typename T>
constexpr T extgcd(const T &a, const T &b, T &x, T &y) {
    T d = a;
    if (b != 0) {
        d = extgcd(b, a % b, y, x);
        y -= (a / b) * x;
    } else {
        x = 1;
        y = 0;
    }
    return d;
}

template <typename M, typename N, class F, ENABLE_IF_T(std::is_integral_v<std::common_type_t<M, N>> and std::is_invocable_r_v<bool, F, std::common_type_t<M, N>>)>
inline constexpr std::common_type_t<M, N> binary_search(const M &ok, const N &ng, F f) {
    std::common_type_t<M, N> _ok = ok, _ng = ng;
    assert(f(_ok));
    while (std::abs(_ok - _ng) > 1) {
        std::common_type_t<M, N> mid = (_ok + _ng) / 2;
        if (f(mid)) {
            _ok = mid;
        } else {
            _ng = mid;
        }
    }
    return _ok;
}

template <typename M, typename N, class F, ENABLE_IF_T(not std::is_integral_v<std::common_type_t<M, N>> and std::is_invocable_r_v<bool, F, std::common_type_t<M, N>>)>
inline constexpr std::common_type_t<M, N> binary_search(const M &ok, const N &ng, F f) {
    std::common_type_t<M, N> _ok = ok, _ng = ng;
    assert(f(_ok));
    for (int i = 0; i < 100; i++) {
        std::common_type_t<M, N> mid = (_ok + _ng) / 2;
        if (f(mid)) {
            _ok = mid;
        } else {
            _ng = mid;
        }
    }
    return _ok;
}

/**
 * 0 <= x < a
 */
inline constexpr bool inrange(long long x, long long a) {
    return 0 <= x and x < a;
}
/**
 * a <= x < b
 */
inline constexpr bool inrange(long long x, long long a, long long b) {
    return a <= x and x < b;
}
/**
 * 0 <= x < a and 0 <= y < b
 */
inline constexpr bool inrect(long long x, long long y, long long a, long long b) {
    return 0 <= x and x < a and 0 <= y and y < b;
}
#line 8 "/home/shogo314/cpp_include/sh-library/base/io.hpp"

namespace tuple_io {
template <typename Tuple, size_t I, typename CharT, typename Traits>
std::basic_istream<CharT, Traits>& read_tuple(std::basic_istream<CharT, Traits>& is, Tuple& t) {
    is >> std::get<I>(t);
    if constexpr (I + 1 < std::tuple_size_v<Tuple>) {
        return read_tuple<Tuple, I + 1>(is, t);
    }
    return is;
}
template <typename Tuple, size_t I, typename CharT, typename Traits>
std::basic_ostream<CharT, Traits>& write_tuple(std::basic_ostream<CharT, Traits>& os, const Tuple& t) {
    os << std::get<I>(t);
    if constexpr (I + 1 < std::tuple_size_v<Tuple>) {
        os << CharT(' ');
        return write_tuple<Tuple, I + 1>(os, t);
    }
    return os;
}
};  // namespace tuple_io

template <typename T1, typename T2, typename CharT, typename Traits>
std::basic_istream<CharT, Traits>& operator>>(std::basic_istream<CharT, Traits>& is, std::pair<T1, T2>& p) {
    is >> p.first >> p.second;
    return is;
}
template <typename... Types, typename CharT, typename Traits>
std::basic_istream<CharT, Traits>& operator>>(std::basic_istream<CharT, Traits>& is, std::tuple<Types...>& p) {
    return tuple_io::read_tuple<std::tuple<Types...>, 0>(is, p);
}
template <typename T, size_t N, typename CharT, typename Traits>
std::basic_istream<CharT, Traits>& operator>>(std::basic_istream<CharT, Traits>& is, std::array<T, N>& a) {
    for (auto& e : a) is >> e;
    return is;
}
template <typename T, typename CharT, typename Traits>
std::basic_istream<CharT, Traits>& operator>>(std::basic_istream<CharT, Traits>& is, std::vector<T>& v) {
    for (auto& e : v) is >> e;
    return is;
}

template <typename T1, typename T2, typename CharT, typename Traits>
std::basic_ostream<CharT, Traits>& operator<<(std::basic_ostream<CharT, Traits>& os, const std::pair<T1, T2>& p) {
    os << p.first << CharT(' ') << p.second;
    return os;
}
template <typename... Types, typename CharT, typename Traits>
std::basic_ostream<CharT, Traits>& operator<<(std::basic_ostream<CharT, Traits>& os, const std::tuple<Types...>& p) {
    return tuple_io::write_tuple<std::tuple<Types...>, 0>(os, p);
}
template <typename T, size_t N, typename CharT, typename Traits>
std::basic_ostream<CharT, Traits>& operator<<(std::basic_ostream<CharT, Traits>& os, const std::array<T, N>& a) {
    for (size_t i = 0; i < N; ++i) {
        if (i) os << CharT(' ');
        os << a[i];
    }
    return os;
}
template <typename T, typename CharT, typename Traits>
std::basic_ostream<CharT, Traits>& operator<<(std::basic_ostream<CharT, Traits>& os, const std::vector<T>& v) {
    for (size_t i = 0; i < v.size(); ++i) {
        if (i) os << CharT(' ');
        os << v[i];
    }
    return os;
}
template <typename T, typename CharT, typename Traits>
std::basic_ostream<CharT, Traits>& operator<<(std::basic_ostream<CharT, Traits>& os, const std::set<T>& s) {
    for (auto itr = s.begin(); itr != s.end(); ++itr) {
        if (itr != s.begin()) os << CharT(' ');
        os << *itr;
    }
    return os;
}

/**
 * @brief 空行出力
 */
void print() { std::cout << '\n'; }
/**
 * @brief 出力して改行
 *
 * @tparam T 型
 * @param x 出力する値
 */
template <typename T>
void print(const T& x) { std::cout << x << '\n'; }
/**
 * @brief 空白区切りで出力して改行
 *
 * @tparam T 1つ目の要素の型
 * @tparam Tail 2つ目以降の要素の型
 * @param x 1つ目の要素
 * @param tail 2つ目以降の要素
 */
template <typename T, typename... Tail>
void print(const T& x, const Tail&... tail) {
    std::cout << x << ' ';
    print(tail...);
}

/**
 * @brief 空行出力
 */
void err() { std::cerr << std::endl; }
/**
 * @brief 出力して改行
 *
 * @tparam T 型
 * @param x 出力する値
 */
template <typename T>
void err(const T& x) { std::cerr << x << std::endl; }
/**
 * @brief 空白区切りで出力して改行
 *
 * @tparam T 1つ目の要素の型
 * @tparam Tail 2つ目以降の要素の型
 * @param x 1つ目の要素
 * @param tail 2つ目以降の要素
 */
template <typename T, typename... Tail>
void err(const T& x, const Tail&... tail) {
    std::cerr << x << ' ';
    err(tail...);
}
#line 3 "/home/shogo314/cpp_include/sh-library/base/type_alias.hpp"

using ll = long long;
using ull = unsigned long long;
using ld = long double;

template <typename T>
using vec = std::vector<T>;
template <typename T, int N>
using ary = std::array<T, N>;
using str = std::string;
using std::deque;
using std::list;
using std::map;
using std::pair;
using std::set;

using pl = pair<ll, ll>;
using pd = pair<ld, ld>;

template <typename T>
using vv = vec<vec<T>>;
template <typename T>
using vvv = vec<vec<vec<T>>>;
using vl = vec<ll>;
using vvl = vv<ll>;
using vvvl = vvv<ll>;
using vs = vec<str>;
using vc = vec<char>;
using vi = vec<int>;
using vb = vec<bool>;

template <typename T1, typename T2>
using vp = vec<pair<T1, T2>>;
using vpl = vec<pl>;
using vvpl = vv<pl>;
using vd = vec<ld>;
using vpd = vec<pd>;

template <int N>
using al = ary<ll, N>;
template <int N1, int N2>
using aal = ary<ary<ll, N2>, N1>;
template <int N>
using val = vec<al<N>>;
template <int N>
using avl = ary<vl,N>;

template <typename T>
using ml = std::map<ll, T>;
using mll = std::map<ll, ll>;
using sl = std::set<ll>;
using spl = set<pl>;
template <int N>
using sal = set<al<N>>;
template <int N>
using asl = ary<sl,N>;

template <typename T>
using heap_max = std::priority_queue<T, std::vector<T>, std::less<T>>;
template <typename T>
using heap_min = std::priority_queue<T, std::vector<T>, std::greater<T>>;
#line 3 "/home/shogo314/cpp_include/sh-library/base/macro.hpp"

#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")

#define all(obj) (obj).begin(), (obj).end()

#define reps(i, a, n) for (long long i = (a); i < (n); i++)
#define rep(i, n) reps(i, 0, (n))
#define rrep(i, n) reps(i, 1, (n) + 1)
#define repds(i, a, n) for (long long i = (n)-1; i >= (a); i--)
#define repd(i, n) repds(i, 0, (n))
#define rrepd(i, n) repds(i, 1, (n) + 1)
#define rep2(i, j, x, y) rep(i, x) rep(j, y)

inline void scan(){}
template<class Head,class... Tail>
inline void scan(Head&head,Tail&... tail){std::cin>>head;scan(tail...);}
#define LL(...) ll __VA_ARGS__;scan(__VA_ARGS__)
#define STR(...) str __VA_ARGS__;scan(__VA_ARGS__)
#define IN(a, x) a x; std::cin >> x;
#define CHAR(x) char x; std::cin >> x;
#define VL(a,n) vl a(n); std::cin >> a;
#define AL(a,k) al<k> a; std::cin >> a;
#define AAL(a,n,m) aal<n,m> a; std::cin >> a;
#define VC(a,n) vc a(n); std::cin >> a;
#define VS(a,n) vs a(n); std::cin >> a;
#define VPL(a,n) vpl a(n); std::cin >> a;
#define VAL(a,n,k) val<k> a(n); std::cin >> a;
#define VVL(a,n,m) vvl a(n,vl(m)); std::cin >> a;
#define SL(a,n) sl a;{VL(b,n);a=sl(all(b));}

#define NO std::cout << "NO" << std::endl; return;
#define YES std::cout << "YES" << std::endl; return;
#define No std::cout << "No" << std::endl; return;
#define Yes std::cout << "Yes" << std::endl; return;
#line 8 "/home/shogo314/cpp_include/sh-library/base/vector_func.hpp"

template <typename T>
std::vector<std::ptrdiff_t> sorted_idx(const std::vector<T> &v) {
    std::vector<std::ptrdiff_t> ret(v.size());
    std::iota(ret.begin(), ret.end(), 0);
    std::sort(ret.begin(), ret.end(), [&](std::ptrdiff_t i, std::ptrdiff_t j) {
        return v[i] < v[j];
    });
    return ret;
}

template <typename T>
inline std::vector<T> &operator++(std::vector<T> &v) {
    for (auto &e : v) e++;
    return v;
}
template <typename T>
inline std::vector<T> operator++(std::vector<T> &v, int) {
    auto res = v;
    for (auto &e : v) e++;
    return res;
}
template <typename T>
inline std::vector<T> &operator--(std::vector<T> &v) {
    for (auto &e : v) e--;
    return v;
}
template <typename T>
inline std::vector<T> operator--(std::vector<T> &v, int) {
    auto res = v;
    for (auto &e : v) e--;
    return res;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> &operator+=(std::vector<T> &v1, const std::vector<U> &v2) {
    if (v2.size() > v1.size()) {
        v1.resize(v2.size());
    }
    for (size_t i = 0; i < v2.size(); i++) {
        v1[i] += v2[i];
    }
    return v1;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> operator+(const std::vector<T> &v1, const std::vector<U> &v2) {
    std::vector<T> res(v1);
    return res += v2;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> &operator+=(std::vector<T> &v, const U &u) {
    for (T &e : v) {
        e += u;
    }
    return v;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> operator+(const std::vector<T> &v, const U &u) {
    std::vector<T> res(v);
    return res += u;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> operator+(const U &u, const std::vector<T> &v) {
    std::vector<T> res(v);
    return res += u;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> &operator*=(std::vector<T> &v1, const std::vector<U> &v2) {
    if (v2.size() > v1.size()) {
        v1.resize(v2.size());
    }
    for (size_t i = 0; i < v2.size(); i++) {
        v1[i] *= v2[i];
    }
    for (size_t i = v2.size(); i < v1.size(); i++) {
        v1[i] *= U(0);
    }
    return v1;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> operator*(const std::vector<T> &v1, const std::vector<U> &v2) {
    std::vector<T> res(v1);
    return res *= v2;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> &operator*=(std::vector<T> &v, const U &u) {
    for (T &e : v) {
        e *= u;
    }
    return v;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> operator*(const std::vector<T> &v, const U &u) {
    std::vector<T> res(v);
    return res *= u;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> operator*(const U &u, const std::vector<T> &v) {
    std::vector<T> res(v);
    return res *= u;
}

template <typename T, typename U>
inline std::vector<T> &assign(std::vector<T> &v1, const std::vector<U> &v2) {
    v1.assign(v2.begin(), v2.end());
    return v1;
}

template <typename T, typename U>
inline std::vector<T> &extend(std::vector<T> &v1, const std::vector<U> &v2) {
    v1.insert(v1.end(), v2.begin(), v2.end());
    return v1;
}

template <typename T, typename U, ENABLE_IF_T(std::is_convertible_v<U, T>)>
inline std::vector<T> &operator|=(std::vector<T> &v1, const std::vector<U> &v2) {
    return extend(v1, v2);
}

template <typename T, typename U, ENABLE_IF_T(std::is_integral_v<U>)>
inline std::vector<T> &operator|=(std::vector<T> &v, const U &u) {
    std::vector<T> w(v);
    v.clear();
    for (int i = 0; i < u; i++) {
        extend(v, w);
    }
    return v;
}

template <typename T, typename U, ENABLE_IF_T(std::is_integral_v<U>)>
inline std::vector<T> operator|(const std::vector<T> &v, const U &u) {
    std::vector<T> res(v);
    return res |= u;
}

template <typename T, typename U, ENABLE_IF_T(std::is_integral_v<U>)>
inline std::vector<T> operator|(const U &u, const std::vector<T> &v) {
    std::vector<T> res(v);
    return res |= u;
}

template <typename T>
inline std::vector<T> abs(const std::vector<T> &v) {
    std::vector<T> ret;
    ret.reserve(v.size());
    for (const T &e : v) ret.push_back(std::abs(e));
    return ret;
}

template <typename T>
std::vector<T> partial_sum(const std::vector<T> &v) {
    std::vector<T> ret(v.size());
    std::partial_sum(v.begin(), v.end(), ret.begin());
    return ret;
}
#line 10 "/home/shogo314/cpp_include/sh-library/py/repr.hpp"

#line 12 "/home/shogo314/cpp_include/sh-library/py/repr.hpp"

namespace detail {

///////////////////////////////////////////////////////////////////
/////////////////////////// declaration ///////////////////////////
///////////////////////////////////////////////////////////////////

inline std::string type_str(const char*);
inline std::string initializer_str(const char*);
inline std::string repr(const char*);

inline std::string type_str(const char&);
inline std::string initializer_str(const char&);
inline std::string repr(const char&);

inline std::string type_str(const int&);
inline std::string initializer_str(const int&);
inline std::string repr(const int&);

inline std::string type_str(const long&);
inline std::string initializer_str(const long&);
inline std::string repr(const long&);

inline std::string type_str(const long long&);
inline std::string initializer_str(const long long&);
inline std::string repr(const long long&);

inline std::string type_str(const unsigned int&);
inline std::string initializer_str(const unsigned int&);
inline std::string repr(const unsigned int&);

inline std::string type_str(const unsigned long&);
inline std::string initializer_str(const unsigned long&);
inline std::string repr(const unsigned long&);

inline std::string type_str(const unsigned long long&);
inline std::string initializer_str(const unsigned long long&);
inline std::string repr(const unsigned long long&);

inline std::string type_str(const std::string&);
inline std::string initializer_str(const std::string&);
inline std::string repr(const std::string&);

inline std::string type_str(const float&);
inline std::string initializer_str(const float&);
inline std::string repr(const float&);

inline std::string type_str(const double&);
inline std::string initializer_str(const double&);
inline std::string repr(const double&);

inline std::string type_str(const long double&);
inline std::string initializer_str(const long double&);
inline std::string repr(const long double&);

template <typename T>
inline std::string type_str(const std::vector<T>&);
template <typename T>
inline std::string initializer_str(const std::vector<T>&);
template <typename T>
inline std::string repr(const std::vector<T>&);

template <typename T>
inline std::string type_str(const std::set<T>&);
template <typename T>
inline std::string initializer_str(const std::set<T>&);
template <typename T>
inline std::string repr(const std::set<T>&);

template <typename T1, typename T2>
inline std::string type_str(const std::pair<T1, T2>&);
template <typename T1, typename T2>
inline std::string initializer_str(const std::pair<T1, T2>&);
template <typename T1, typename T2>
inline std::string repr(const std::pair<T1, T2>&);

template <typename T1, typename T2>
inline std::string type_str(const std::map<T1, T2>&);
template <typename T1, typename T2>
inline std::string initializer_str(const std::map<T1, T2>&);
template <typename T1, typename T2>
inline std::string repr(const std::map<T1, T2>&);

template <typename T, size_t N>
inline std::string type_str(const std::array<T, N>&);
template <typename T, size_t N>
inline std::string initializer_str(const std::array<T, N>&);
template <typename T, size_t N>
inline std::string repr(const std::array<T, N>&);

///////////////////////////////////////////////////////////////////
/////////////////////////// definition ////////////////////////////
///////////////////////////////////////////////////////////////////

template <typename T, std::enable_if_t<not has_type_str_v<T>, std::nullptr_t> = nullptr>
inline std::string type_str(const T& t) {
    return "T";
}
template <typename T, std::enable_if_t<has_type_str_v<T>, std::nullptr_t> = nullptr>
inline std::string type_str(const T& t) {
    return t.type_str();
}
template <typename T, std::enable_if_t<not has_initializer_str_v<T>, std::nullptr_t> = nullptr>
inline std::string initializer_str(const T& t) {
    return "{}";
}
template <typename T, std::enable_if_t<has_initializer_str_v<T>, std::nullptr_t> = nullptr>
inline std::string initializer_str(const T& t) {
    return t.initializer_str();
}
template <typename T, std::enable_if_t<not has_repr_v<T>, std::nullptr_t> = nullptr>
inline std::string repr(const T& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}
template <typename T, std::enable_if_t<has_repr_v<T>, std::nullptr_t> = nullptr>
inline std::string repr(const T& t) {
    return t.repr();
}

inline std::string type_str(const char*) {
    return "char*";
}
inline std::string initializer_str(const char* t) {
    return "\"" + std::string(t) + "\"";
}
inline std::string repr(const char* t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const char&) {
    return "char";
}
inline std::string initializer_str(const char& t) {
    std::vector<std::string> v(128, "");
    v['\0'] = "\\0";
    v['\a'] = "\\a";
    v['\b'] = "\\b";
    v['\f'] = "\\f";
    v['\r'] = "\\r";
    v['\n'] = "\\n";
    v['\v'] = "\\v";
    v['\''] = "\\'";
    v['\\'] = "\\\\";
    if (!v[t].empty())
        return "'" + v[t] + "'";
    if (32 <= t and t <= 126)
        return "'" + std::string{t} + "'";
    return "'\\" + std::to_string(t / 64) + std::to_string(t / 8 % 8) + std::to_string(t % 8) + "'";
}
inline std::string repr(const char& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const int&) {
    return "int";
}
inline std::string initializer_str(const int& t) {
    return std::to_string(t);
}
inline std::string repr(const int& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const long&) {
    return "long";
}
inline std::string initializer_str(const long& t) {
    return std::to_string(t) + "l";
}
inline std::string repr(const long& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const long long&) {
    return "long long";
}
inline std::string initializer_str(const long long& t) {
    return std::to_string(t) + "ll";
}
inline std::string repr(const long long& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const unsigned int&) {
    return "unsigned int";
}
inline std::string initializer_str(const unsigned int& t) {
    return std::to_string(t) + "u";
}
inline std::string repr(const unsigned int& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const unsigned long&) {
    return "unsigned long";
}
inline std::string initializer_str(const unsigned long& t) {
    return std::to_string(t) + "ul";
}
inline std::string repr(const unsigned long& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const unsigned long long&) {
    return "unsigned long long";
}
inline std::string initializer_str(const unsigned long long& t) {
    return std::to_string(t) + "ull";
}
inline std::string repr(const unsigned long long& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const std::string&) {
    return "std::string";
}
inline std::string initializer_str(const std::string& t) {
    return "\"" + t + "\"";
}
inline std::string repr(const std::string& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}

inline std::string type_str(const float&) {
    return "float";
}
inline std::string initializer_str(const float& t) {
    return std::to_string(t) + "f";
}
inline std::string repr(const float& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const double&) {
    return "double";
}
inline std::string initializer_str(const double& t) {
    return std::to_string(t);
}
inline std::string repr(const double& t) {
    return detail::initializer_str(t);
}

inline std::string type_str(const long double&) {
    return "long double";
}
inline std::string initializer_str(const long double& t) {
    return std::to_string(t) + "l";
}
inline std::string repr(const long double& t) {
    return detail::initializer_str(t);
}

template <typename T>
inline std::string type_str(const std::vector<T>&) {
    T t;
    return "std::vector<" + detail::type_str(t) + ">";
}
template <typename T>
inline std::string initializer_str(const std::vector<T>& t) {
    std::string ret;
    ret += "{";
    for (auto itr = t.begin(); itr != t.end(); itr++) {
        if (itr != t.begin()) ret += ", ";
        ret += detail::initializer_str(*itr);
    }
    ret += "}";
    return ret;
}
template <typename T>
inline std::string repr(const std::vector<T>& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}

template <typename T>
inline std::string type_str(const std::set<T>&) {
    T t;
    return "std::set<" + detail::type_str(t) + ">";
}
template <typename T>
inline std::string initializer_str(const std::set<T>& t) {
    std::string ret;
    ret += "{";
    for (auto itr = t.begin(); itr != t.end(); itr++) {
        if (itr != t.begin()) ret += ", ";
        ret += detail::initializer_str(*itr);
    }
    ret += "}";
    return ret;
}
template <typename T>
inline std::string repr(const std::set<T>& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}

template <typename T1, typename T2>
inline std::string type_str(const std::pair<T1, T2>&) {
    T1 t1;
    T2 t2;
    return "std::pair<" + detail::type_str(t1) + ", " + detail::type_str(t2) + ">";
}
template <typename T1, typename T2>
inline std::string initializer_str(const std::pair<T1, T2>& t) {
    return "{" + detail::repr(t.first) + ", " + detail::repr(t.second) + "}";
}
template <typename T1, typename T2>
inline std::string repr(const std::pair<T1, T2>& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}

template <typename T1, typename T2>
inline std::string type_str(const std::map<T1, T2>&) {
    T1 t1;
    T2 t2;
    return "std::map<" + detail::type_str(t1) + ", " + detail::type_str(t2) + ">";
}
template <typename T1, typename T2>
inline std::string initializer_str(const std::map<T1, T2>& t) {
    std::string ret;
    ret += "{";
    for (auto itr = t.begin(); itr != t.end(); itr++) {
        if (itr != t.begin()) ret += ", ";
        ret += "{";
        ret += detail::initializer_str(itr->first);
        ret += ", ";
        ret += detail::initializer_str(itr->second);
        ret += "}";
    }
    ret += "}";
    return ret;
}
template <typename T1, typename T2>
inline std::string repr(const std::map<T1, T2>& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}

template <typename T, size_t N>
inline std::string type_str(const std::array<T, N>&) {
    T t;
    return "std::array<" + detail::type_str(t) + ", " + std::to_string(N) + ">";
}
template <typename T, size_t N>
inline std::string initializer_str(const std::array<T, N>& t) {
    std::string ret;
    ret += "{";
    for (auto itr = t.begin(); itr != t.end(); itr++) {
        if (itr != t.begin()) ret += ", ";
        ret += detail::initializer_str(*itr);
    }
    ret += "}";
    return ret;
}
template <typename T, size_t N>
inline std::string repr(const std::array<T, N>& t) {
    return detail::type_str(t) + detail::initializer_str(t);
}

}  // namespace detail

template <typename T>
inline std::string type_str(const T& t) {
    return detail::type_str(t);
}
template <typename T>
inline std::string initializer_str(const T& t) {
    return detail::initializer_str(t);
}
template <typename T>
inline std::string repr(const T& t) {
    return detail::repr(t);
}

#define DEBUG(x) { std::cerr << #x << " = " << repr(x) << std::endl; }
#line 4 "main.cpp"

void solve() {
    LL(H, W, N, D);
    VAL(XY, N, 2);
    UnionFind uf(N);
    vvl g(H, vl(W, -1));
    rep(i, N) {
        auto& [X, Y] = XY[i];
        X--;
        Y--;
        g[X][Y] = i;
    }
    rep(i, N) {
        auto [X, Y] = XY[i];
        reps(dx, -D, D + 1) reps(dy, -D, D + 1) {
            if (abs(dx) + abs(dy) > D) {
                continue;
            }
            if (dx == 0 and dy == 0) continue;
            if (not inrect(X + dx, Y + dy, H, W)) continue;
            if (g[X + dx][Y + dy] >= 0) {
                uf.merge(i, g[X + dx][Y + dy]);
            }
        }
    }
    ll min_ans, max_ans, ans = 0;
    for (const auto& a : uf.groups()) {
        if (a.size() > 1) ans++;
    }
    min_ans = max_ans = ans;
    rep2(x, y, H, W) {
        if (g[x][y] >= 0) continue;
        sl s;
        ll tmp = 0;
        reps(dx, -D, D + 1) reps(dy, -D, D + 1) {
            if (abs(dx) + abs(dy) > D) {
                continue;
            }
            if (dx == 0 and dy == 0) continue;
            if (not inrect(x + dx, y + dy, H, W)) continue;
            if (g[x + dx][y + dy] >= 0) {
                s.insert(uf.leader(g[x + dx][y + dy]));
                if (uf.size(g[x + dx][y + dy]) == 1) tmp++;
            }
        }
        if (s.empty()) continue;
        tmp = 1 - (s.size() - tmp);
        chmin(min_ans, ans + tmp);
        chmax(max_ans, ans + tmp);
    }
    print(min_ans, max_ans);
}

int main() {
    std::cin.tie(nullptr);
    std::ios_base::sync_with_stdio(false);
    solve();
}