#line 2 "/home/yuruhiya/programming/library/Utility/get_MOD.cpp"
constexpr long long get_MOD() {
#ifdef SET_MOD
	return SET_MOD;
#else
	return 1000000007;
#endif
}
#line 3 "/home/yuruhiya/programming/library/Utility/constants.cpp"
#include <vector>
#include <string>
#include <utility>
#include <queue>

#define rep(i, n) for (int i = 0; i < (n); ++i)
#define FOR(i, m, n) for (int i = (m); i < (n); ++i)
#define rrep(i, n) for (int i = (n)-1; i >= 0; --i)
#define rfor(i, m, n) for (int i = (m); i >= (n); --i)
#define INTERNAL_CAT_IMPL(s1, s2) s1##s2
#define INTERNAL_CAT(s1, s2) INTERNAL_CAT_IMPL(s1, s2)
#ifdef __COUNTER__
#define loop(n) rep(INTERNAL_CAT(_i, __COUNTER__), n)
#else
#define loop(n) rep(INTERNAL_CAT(_i, __COUNTER__), n)
#endif
#define unless(c) if (!(c))
#define ALL(x) (x).begin(), (x).end()
#define RALL(x) (x).rbegin(), (x).rend()
#define range_it(a, l, r) (a).begin() + (l), (a).begin() + (r)

using ll = long long;
using LD = long double;
using VB = std::vector<bool>;
using VVB = std::vector<VB>;
using VI = std::vector<int>;
using VVI = std::vector<VI>;
using VL = std::vector<ll>;
using VVL = std::vector<VL>;
using VS = std::vector<std::string>;
using VD = std::vector<LD>;
using PII = std::pair<int, int>;
using VP = std::vector<PII>;
using PLL = std::pair<ll, ll>;
using VPL = std::vector<PLL>;
template <class T> using PQ = std::priority_queue<T>;
template <class T> using PQS = std::priority_queue<T, std::vector<T>, std::greater<T>>;

constexpr int inf = 1000000000;
constexpr long long inf_ll = 1000000000000000000ll, MOD = get_MOD();
constexpr long double PI = 3.14159265358979323846, tau = PI * 2, EPS = 1e-12;
#line 2 "/home/yuruhiya/programming/library/Utility/Scanner.cpp"
#include <iostream>
#line 6 "/home/yuruhiya/programming/library/Utility/Scanner.cpp"
#include <tuple>
#include <type_traits>

#ifdef _WIN32
#define getchar_unlocked _getchar_nolock
#define putchar_unlocked _putchar_nolock
#define fwrite_unlocked fwrite
#define fflush_unlocked fflush
#endif
class Scanner {
	template <class T, class = void> struct has_scan : std::false_type {};
	template <class T>
	struct has_scan<T, std::void_t<decltype(std::declval<T>().template scan<Scanner>())>>
	    : std::true_type {};

public:
	static int gc() {
		return getchar_unlocked();
	}
	static char next_char() {
		char c;
		scan(c);
		return c;
	}
	template <class T> static void scan(T& v) {
		if (has_scan<T>::value) {
			v.template scan<Scanner>();
		} else {
			std::cin >> v;
		}
	}
	static void scan(char& v) {
		while (std::isspace(v = gc()))
			;
	}
	static void scan(bool& v) {
		v = next_char() != '0';
	}
	static void scan(std::vector<bool>::reference v) {
		bool b;
		scan(b);
		v = b;
	}
	static void scan(std::string& v) {
		v.clear();
		for (char c = next_char(); !std::isspace(c); c = gc()) v += c;
	}
	static void scan(int& v) {
		v = 0;
		bool neg = false;
		char c = next_char();
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; std::isdigit(c); c = gc()) v = v * 10 + (c - '0');
		if (neg) v = -v;
	}
	static void scan(long long& v) {
		v = 0;
		bool neg = false;
		char c = next_char();
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; std::isdigit(c); c = gc()) v = v * 10 + (c - '0');
		if (neg) v = -v;
	}
	static void scan(double& v) {
		v = 0;
		double dp = 1;
		bool neg = false, after_dp = false;
		char c = next_char();
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; std::isdigit(c) || c == '.'; c = gc()) {
			if (c == '.') {
				after_dp = true;
			} else if (after_dp) {
				v += (c - '0') * (dp *= 0.1);
			} else {
				v = v * 10 + (c - '0');
			}
		}
		if (neg) v = -v;
	}
	static void scan(long double& v) {
		v = 0;
		long double dp = 1;
		bool neg = false, after_dp = false;
		char c = next_char();
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; std::isdigit(c) || c == '.'; c = gc()) {
			if (c == '.') {
				after_dp = true;
			} else if (after_dp) {
				v += (c - '0') * (dp *= 0.1);
			} else {
				v = v * 10 + (c - '0');
			}
		}
		if (neg) v = -v;
	}
	template <class T, class U> static void scan(std::pair<T, U>& v) {
		scan(v.first);
		scan(v.second);
	}
	template <class T, std::enable_if_t<!std::is_same_v<bool, T>, std::nullptr_t> = nullptr>
	static void scan(std::vector<T>& v) {
		for (auto& e : v) scan(e);
	}
	template <class T, std::enable_if_t<std::is_same_v<bool, T>, std::nullptr_t> = nullptr>
	static void scan(std::vector<T>& v) {
		for (auto e : v) scan(e);
	}

private:
	template <std::size_t N = 0, class T> static void scan_tuple_impl(T& v) {
		if constexpr (N < std::tuple_size_v<T>) {
			scan(std::get<N>(v));
			scan_tuple_impl<N + 1>(v);
		}
	}

public:
	template <class... T> static void scan(std::tuple<T...>& v) {
		scan_tuple_impl(v);
	}

private:
	struct Read2DVectorHelper {
		std::size_t h, w;
		Read2DVectorHelper(std::size_t _h, std::size_t _w) : h(_h), w(_w) {}
		template <class T> operator std::vector<std::vector<T>>() {
			std::vector vector(h, std::vector<T>(w));
			scan(vector);
			return vector;
		}
	};
	struct ReadVectorHelper {
		std::size_t n;
		ReadVectorHelper(std::size_t _n) : n(_n) {}
		template <class T> operator std::vector<T>() {
			std::vector<T> vector(n);
			scan(vector);
			return vector;
		}
		auto operator[](std::size_t m) {
			return Read2DVectorHelper(n, m);
		}
	};

public:
	template <class T> T read() const {
		T result;
		scan(result);
		return result;
	}
	template <class T> auto read(std::size_t n) const {
		std::vector<T> result(n);
		scan(result);
		return result;
	}
	template <class T> auto read(std::size_t h, std::size_t w) const {
		std::vector result(h, std::vector<T>(w));
		scan(result);
		return result;
	}
	std::string read_line() const {
		std::string v;
		for (char c = gc(); c != '\n' && c != '\0'; c = gc()) v += c;
		return v;
	}
	template <class T> operator T() const {
		return read<T>();
	}
	int operator--(int) const {
		return read<int>() - 1;
	}
	auto operator[](std::size_t n) const {
		return ReadVectorHelper(n);
	}
	auto operator[](const std::pair<std::size_t, std::size_t>& nm) const {
		return Read2DVectorHelper(nm.first, nm.second);
	}
	void operator()() const {}
	template <class H, class... T> void operator()(H&& h, T&&... t) const {
		scan(h);
		operator()(std::forward<T>(t)...);
	}

private:
	template <template <class...> class, class...> struct Column;
	template <template <class...> class V, class Head, class... Tail>
	struct Column<V, Head, Tail...> {
		template <class... Args> using vec = V<std::vector<Head>, Args...>;
		using type = typename Column<vec, Tail...>::type;
	};
	template <template <class...> class V> struct Column<V> { using type = V<>; };
	template <class... T> using column_t = typename Column<std::tuple, T...>::type;
	template <std::size_t N = 0, class T> void column_impl(T& t) const {
		if constexpr (N < std::tuple_size_v<T>) {
			auto& vec = std::get<N>(t);
			using V = typename std::remove_reference_t<decltype(vec)>::value_type;
			vec.push_back(read<V>());
			column_impl<N + 1>(t);
		}
	}

public:
	template <class... T> auto column(std::size_t h) const {
		column_t<T...> result;
		while (h--) column_impl(result);
		return result;
	}
} in;
#define inputs(T, ...) \
	T __VA_ARGS__;     \
	in(__VA_ARGS__)
#define ini(...) inputs(int, __VA_ARGS__)
#define inl(...) inputs(long long, __VA_ARGS__)
#define ins(...) inputs(std::string, __VA_ARGS__)
#line 5 "/home/yuruhiya/programming/library/Utility/Printer.cpp"
#include <array>
#line 7 "/home/yuruhiya/programming/library/Utility/Printer.cpp"
#include <string_view>
#include <optional>
#include <charconv>
#line 11 "/home/yuruhiya/programming/library/Utility/Printer.cpp"
#include <cstring>
#include <cassert>

class Printer {
public:
	struct BoolString {
		std::string_view t, f;
		BoolString(std::string_view _t, std::string_view _f) : t(_t), f(_f) {}
	};
	struct Separator {
		std::string_view div, sep, last;
		Separator(std::string_view _div, std::string_view _sep, std::string_view _last)
		    : div(_div), sep(_sep), last(_last) {}
	};

	inline static const BoolString Yes{"Yes", "No"}, yes{"yes", "no"}, YES{"YES", "NO"},
	    Int{"1", "0"}, Possible{"Possible", "Impossible"};
	inline static const Separator space{" ", " ", "\n"}, no_space{"", "", "\n"},
	    endl{"\n", "\n", "\n"}, comma{",", ",", "\n"}, no_endl{" ", " ", ""},
	    sep_endl{" ", "\n", "\n"};

	BoolString bool_str{Yes};
	Separator separator{space};

private:
	template <class T, class = void> struct has_print : std::false_type {};
	template <class T>
	struct has_print<T,
	                 std::void_t<decltype(std::declval<T>().print(std::declval<Printer>()))>>
	    : std::true_type {};

public:
	void print(int v) const {
		char buf[12]{};
		if (auto [ptr, e] = std::to_chars(std::begin(buf), std::end(buf), v);
		    e == std::errc{}) {
			print(std::string_view(buf, ptr - buf));
		} else {
			assert(false);
		}
	}
	void print(long long v) const {
		char buf[21]{};
		if (auto [ptr, e] = std::to_chars(std::begin(buf), std::end(buf), v);
		    e == std::errc{}) {
			print(std::string_view(buf, ptr - buf));
		} else {
			assert(false);
		}
	}
	void print(bool v) const {
		print(v ? bool_str.t : bool_str.f);
	}
	void print(std::vector<bool>::reference v) const {
		print(v ? bool_str.t : bool_str.f);
	}
	void print(char v) const {
		putchar_unlocked(v);
	}
	void print(std::string_view v) const {
		fwrite_unlocked(v.data(), sizeof(std::string_view::value_type), v.size(), stdout);
	}
	void print(double v) const {
		std::printf("%.20f", v);
	}
	void print(long double v) const {
		std::printf("%.20Lf", v);
	}
	template <class T, std::enable_if_t<has_print<T>::value, std::nullptr_t> = nullptr>
	void print(const T& v) const {
		v.print(*this);
	}
	template <class T, std::enable_if_t<!has_print<T>::value, std::nullptr_t> = nullptr>
	void print(const T& v) const {
		std::cout << v;
	}
	template <class T, class U> void print(const std::pair<T, U>& v) const {
		print(v.first);
		print(separator.div);
		print(v.second);
	}
	template <class T> void print(const std::optional<T>& v) const {
		print(*v);
	}
	template <class InputIterater>
	void print_range(const InputIterater& begin, const InputIterater& end) const {
		for (InputIterater i = begin; i != end; ++i) {
			if (i != begin) print(separator.sep);
			print(*i);
		}
	}
	template <class T> void print(const std::vector<T>& v) const {
		print_range(v.begin(), v.end());
	}
	template <class T, std::size_t N> void print(const std::array<T, N>& v) const {
		print_range(v.begin(), v.end());
	}
	template <class T> void print(const std::vector<std::vector<T>>& v) const {
		for (std::size_t i = 0; i < v.size(); ++i) {
			if (i) print(separator.last);
			print(v[i]);
		}
	}

	Printer() = default;
	Printer(const BoolString& _bool_str, const Separator& _separator)
	    : bool_str(_bool_str), separator(_separator) {}
	Printer& operator()() {
		print(separator.last);
		return *this;
	}
	template <class Head> Printer& operator()(Head&& head) {
		print(head);
		print(separator.last);
		return *this;
	}
	template <class Head, class... Tail> Printer& operator()(Head&& head, Tail&&... tail) {
		print(head);
		print(separator.sep);
		return operator()(std::forward<Tail>(tail)...);
	}
	template <class... Args> Printer& flag(bool f, Args&&... args) {
		if (f) {
			return operator()(std::forward<Args>(args)...);
		} else {
			return *this;
		}
	}
	template <class InputIterator>
	Printer& range(const InputIterator& begin, const InputIterator& end) {
		print_range(begin, end);
		print(separator.last);
		return *this;
	}
	template <class Container> Printer& range(const Container& a) {
		range(a.begin(), a.end());
		return *this;
	}
	template <class... T> void exit(T&&... t) {
		operator()(std::forward<T>(t)...);
		std::exit(EXIT_SUCCESS);
	}
	Printer& flush() {
		fflush_unlocked(stdout);
		return *this;
	}
	Printer& set(const BoolString& _bool_str) {
		bool_str = _bool_str;
		return *this;
	}
	Printer& set(const Separator& _separator) {
		separator = _separator;
		return *this;
	}
	Printer& set(std::string_view t, std::string_view f) {
		bool_str = BoolString(t, f);
		return *this;
	}
} out;
#line 3 "/home/yuruhiya/programming/library/Utility/Step.cpp"
#include <iterator>
#include <algorithm>
#line 6 "/home/yuruhiya/programming/library/Utility/Step.cpp"

template <class T> class step_iterator {
public:
	using value_type = T;
	using difference_type = T;
	using iterator_category = std::random_access_iterator_tag;
	using reference = T&;
	using pointer = T*;

private:
	value_type start_m, size_m, step_m, index_m;

public:
	constexpr step_iterator()
	    : start_m(value_type()), size_m(value_type()), step_m(value_type()), index_m(0) {}
	constexpr step_iterator(value_type _start, value_type _size, value_type _step)
	    : start_m(_start), size_m(_size), step_m(_step), index_m(0) {}
	value_type operator*() const noexcept {
		return value();
	}
	step_iterator& operator++() noexcept {
		++index_m;
		return *this;
	}
	step_iterator& operator++(int) noexcept {
		auto tmp = *this;
		++*this;
		return tmp;
	}
	step_iterator& operator--() noexcept {
		--index_m;
		return *this;
	}
	step_iterator& operator--(int) noexcept {
		auto tmp = *this;
		--*this;
		return tmp;
	}
	step_iterator& operator+=(difference_type n) {
		index_m += n;
		return *this;
	}
	step_iterator operator+(difference_type n) const {
		return step_iterator(*this) += n;
	}
	friend step_iterator operator+(difference_type n, step_iterator i) {
		return i + n;
	}
	step_iterator& operator-=(difference_type n) {
		index_m -= n;
		return *this;
	}
	step_iterator operator-(difference_type n) const {
		return step_iterator(*this) -= n;
	}
	friend step_iterator operator-(difference_type n, step_iterator i) {
		return i - n;
	}
	difference_type operator-(const step_iterator& other) const {
		assert(start_m == other.start_m);
		assert(size_m == other.size_m);
		assert(step_m == other.step_m);
		return index_m - other.index_m;
	}
	bool operator==(const step_iterator& other) const noexcept {
		return value() == other.value();
	}
	bool operator!=(const step_iterator& other) const noexcept {
		return value() != other.value();
	}
	bool operator<(const step_iterator& other) const noexcept {
		return value() < other.value();
	}
	bool operator<=(const step_iterator& other) const noexcept {
		return value() <= other.value();
	}
	bool operator>(const step_iterator& other) const noexcept {
		return value() > other.value();
	}
	bool operator>=(const step_iterator& other) const noexcept {
		return value() >= other.value();
	}
	constexpr value_type value() const noexcept {
		return start_m + step_m * index_m;
	}
};

template <class T> class Step {
public:
	using value_type = T;
	using iterator = step_iterator<value_type>;

private:
	value_type start_m, size_m, step_m;

public:
	constexpr Step(value_type _start, value_type _size, value_type _step)
	    : start_m(_start), size_m(std::max<value_type>(0, _size)), step_m(_step) {}
	constexpr iterator begin() const {
		return iterator(start_m, size_m, step_m);
	}
	constexpr iterator end() const {
		return iterator(start_m, size_m, step_m) + size_m;
	}
	constexpr value_type start() const {
		return start_m;
	}
	constexpr value_type size() const {
		return size_m;
	}
	constexpr value_type step() const {
		return step_m;
	}
	constexpr value_type sum() const {
		return start() * size() + step() * (size() * (size() - 1) / 2);
	}
	operator std::vector<value_type>() const {
		return to_a();
	}
	auto to_a() const {
		std::vector<value_type> result;
		result.reserve(size());
		for (auto i : *this) {
			result.push_back(i);
		}
		return result;
	}
};
template <class T> constexpr auto upto(T from, T to, bool exclusive = true) {
	return Step<T>(from, to - from + exclusive, 1);
}
template <class T> constexpr auto downto(T from, T to, bool exclusive = true) {
	return Step<T>(from, from - to + exclusive, -1);
}
template <class T> constexpr auto times(T n, bool exclusive = false) {
	return Step<T>(0, n + static_cast<T>(exclusive), 1);
}
#line 4 "/home/yuruhiya/programming/library/Utility/Ruby.cpp"
#include <map>
#line 6 "/home/yuruhiya/programming/library/Utility/Ruby.cpp"
#include <numeric>
#line 9 "/home/yuruhiya/programming/library/Utility/Ruby.cpp"

template <class F> struct Callable {
	F func;
	Callable(const F& f) : func(f) {}
};
template <class T, class F> auto operator|(const T& v, const Callable<F>& c) {
	return c.func(v);
}

struct Sort_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			std::sort(std::begin(v), std::end(v), f);
			return v;
		});
	}
	template <class T> friend auto operator|(T v, [[maybe_unused]] const Sort_impl& c) {
		std::sort(std::begin(v), std::end(v));
		return v;
	}
} Sort;
struct SortBy_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			std::sort(std::begin(v), std::end(v),
			          [&](const auto& i, const auto& j) { return f(i) < f(j); });
			return v;
		});
	}
} SortBy;
struct RSort_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			std::sort(rbegin(v), rend(v), f);
			return v;
		});
	}
	template <class T> friend auto operator|(T v, [[maybe_unused]] const RSort_impl& c) {
		std::sort(rbegin(v), rend(v));
		return v;
	}
} RSort;
struct RSortBy_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			std::sort(std::begin(v), std::end(v),
			          [&](const auto& i, const auto& j) { return f(i) > f(j); });
			return v;
		});
	}
} RSortBy;
struct Reverse_impl {
	template <class T> friend auto operator|(T v, const Reverse_impl& c) {
		std::reverse(std::begin(v), std::end(v));
		return v;
	}
} Reverse;
struct Unique_impl {
	template <class T> friend auto operator|(T v, const Unique_impl& c) {
		v.erase(std::unique(std::begin(v), std::end(v), std::end(v)));
		return v;
	}
	template <class T, class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			v.erase(std::unique(std::begin(v), std::end(v), f), std::end(v));
			return v;
		});
	}
} Unique;
struct Uniq_impl {
	template <class T> friend auto operator|(T v, const Uniq_impl& c) {
		std::sort(std::begin(v), std::end(v));
		v.erase(std::unique(std::begin(v), std::end(v)), std::end(v));
		return v;
	}
} Uniq;
struct Rotate_impl {
	auto operator()(int&& left) {
		return Callable([&](auto v) {
			int s = static_cast<int>(std::size(v));
			assert(-s <= left && left <= s);
			if (0 <= left) {
				std::rotate(std::begin(v), std::begin(v) + left, std::end(v));
			} else {
				std::rotate(std::begin(v), std::end(v) + left, std::end(v));
			}
			return v;
		});
	}
} Rotate;
struct Max_impl {
	template <class F> auto operator()(F&& f) {
		return Callable(
		    [&](auto v) { return *std::max_element(std::begin(v), std::end(v), f); });
	}
	template <class T> friend auto operator|(T v, const Max_impl& c) {
		return *std::max_element(std::begin(v), std::end(v));
	}
} Max;
struct Min_impl {
	template <class F> auto operator()(F&& f) {
		return Callable(
		    [&](auto v) { return *std::min_element(std::begin(v), std::end(v), f); });
	}
	template <class T> friend auto operator|(T v, const Min_impl& c) {
		return *std::min_element(std::begin(v), std::end(v));
	}
} Min;
struct MaxPos_impl {
	template <class T> friend auto operator|(T v, const MaxPos_impl& c) {
		return std::max_element(std::begin(v), std::end(v)) - std::begin(v);
	}
} MaxPos;
struct MinPos_impl {
	template <class T> friend auto operator|(T v, const MinPos_impl& c) {
		return std::min_element(std::begin(v), std::end(v)) - std::begin(v);
	}
} MinPos;
struct MaxBy_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			auto max_it = std::begin(v);
			auto max_val = f(*max_it);
			for (auto it = std::next(std::begin(v)); it != std::end(v); ++it) {
				if (auto val = f(*it); max_val < val) {
					max_it = it;
					max_val = val;
				}
			}
			return *max_it;
		});
	}
} MaxBy;
struct MinBy_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			auto min_it = std::begin(v);
			auto min_val = f(*min_it);
			for (auto it = std::next(std::begin(v)); it != std::end(v); ++it) {
				if (auto val = f(*it); min_val > val) {
					min_it = it;
					min_val = val;
				}
			}
			return *min_it;
		});
	}
} MinBy;
struct MaxOf_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			auto max_val = f(*std::begin(v));
			for (auto it = std::next(std::begin(v)); it != std::end(v); ++it) {
				if (auto val = f(*it); max_val < val) {
					max_val = val;
				}
			}
			return max_val;
		});
	}
} MaxOf;
struct MinOf_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			auto min_val = f(*std::begin(v));
			for (auto it = std::next(std::begin(v)); it != std::end(v); ++it) {
				if (auto val = f(*it); min_val > val) {
					min_val = val;
				}
			}
			return min_val;
		});
	}
} MinOf;
struct Count_impl {
	template <class V> auto operator()(const V& val) {
		return Callable([&](auto v) { return std::count(std::begin(v), std::end(v), val); });
	}
} Count;
struct CountIf_impl {
	template <class F> auto operator()(const F& f) {
		return Callable([&](auto v) { return std::count_if(std::begin(v), std::end(v), f); });
	}
} CountIf;
struct Index_impl {
	template <class V> auto operator()(const V& val) {
		return Callable([&](auto v) -> std::optional<int> {
			auto result = std::find(std::begin(v), std::end(v), val);
			return result != std::end(v) ? std::optional(result - std::begin(v))
			                             : std::nullopt;
		});
	}
	template <class V> auto operator()(const V& val, std::size_t i) {
		return Callable([&](auto v) -> std::optional<int> {
			auto result = std::find(std::next(std::begin(v), i), std::end(v), val);
			return result != std::end(v) ? std::optional(result - std::begin(v))
			                             : std::nullopt;
		});
	}
} Index;
struct IndexIf_impl {
	template <class F> auto operator()(const F& f) {
		return Callable([&](auto v) -> std::optional<int> {
			auto result = std::find_if(std::begin(v), std::end(v), f);
			return result != std::end(v) ? std::optional(result - std::begin(v))
			                             : std::nullopt;
		});
	}
} IndexIf;
struct FindIf_impl {
	template <class F> auto operator()(const F& f) {
		return Callable([&](auto v) -> std::optional<typename decltype(v)::value_type> {
			auto result = std::find_if(std::begin(v), std::end(v), f);
			return result != std::end(v) ? std::optional(*result) : std::nullopt;
		});
	}
} FindIf;
struct Sum_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			return std::accumulate(std::next(std::begin(v)), std::end(v), f(*std::begin(v)),
			                       [&](const auto& a, const auto& b) { return a + f(b); });
		});
	}
	template <class T> friend auto operator|(T v, [[maybe_unused]] const Sum_impl& c) {
		return std::accumulate(std::begin(v), std::end(v), typename T::value_type{});
	}
} Sum;
struct Includes {
	template <class V> auto operator()(const V& val) {
		return Callable(
		    [&](auto v) { return std::find(std::begin(v), std::end(v), val) != std::end(v); });
	}
} Includes;
struct IncludesIf_impl {
	template <class F> auto operator()(const F& f) {
		return Callable([&](auto v) {
			return std::find_if(std::begin(v), std::end(v), f) != std::end(v);
		});
	}
} IncludesIf;
struct RemoveIf_impl {
	template <class F> auto operator()(const F& f) {
		return Callable([&](auto v) {
			v.erase(std::remove_if(std::begin(v), std::end(v), f), std::end(v));
			return v;
		});
	}
} RemoveIf;
struct Each_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			for (const auto& i : v) {
				f(i);
			}
		});
	}
} Each;
struct EachConsPair_impl {
	template <class T, class value_type = typename T::value_type>
	friend auto operator|(const T& v, EachConsPair_impl& c) {
		std::vector<std::pair<value_type, value_type>> result;
		if (std::size(v) >= 2) {
			result.reserve(std::size(v) - 1);
			for (std::size_t i = 0; i < std::size(v) - 1; ++i) {
				result.emplace_back(v[i], v[i + 1]);
			}
		}
		return result;
	}
} EachConsPair;
struct Select_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			using value_type = typename decltype(v)::value_type;
			std::vector<value_type> result;
			for (const auto& i : v) {
				if (f(i)) result.push_back(i);
			}
			return result;
		});
	}
} Select;
struct Map_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			using result_type = std::invoke_result_t<F, typename decltype(v)::value_type>;
			std::vector<result_type> result;
			result.reserve(std::size(v));
			for (const auto& i : v) {
				result.push_back(f(i));
			}
			return result;
		});
	}
} Map;
struct Indexed_impl {
	template <class T> friend auto operator|(const T& v, Indexed_impl& c) {
		using value_type = typename T::value_type;
		std::vector<std::pair<value_type, int>> result;
		result.reserve(std::size(v));
		int index = 0;
		for (const auto& i : v) {
			result.emplace_back(i, index++);
		}
		return result;
	}
} Indexed;
struct AllOf_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			for (const auto& i : v) {
				if (!f(i)) return false;
			}
			return true;
		});
	}
} AllOf;
struct AnyOf_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			for (const auto& i : v) {
				if (f(i)) return true;
			}
			return false;
		});
	}
} AnyOf;
struct NoneOf_impl {
	template <class F> auto operator()(F&& f) {
		return Callable([&](auto v) {
			for (const auto& i : v) {
				if (f(i)) return false;
			}
			return true;
		});
	}
} NoneOf;

struct Tally_impl {
	auto operator()(std::size_t max_val) {
		return Callable([&](auto v) {
			std::vector<std::size_t> result(max_val);
			for (const auto& i : v) {
				result[static_cast<std::size_t>(i)]++;
			}
			return result;
		});
	}
	template <class T, class value_type = typename T::value_type>
	friend auto operator|(const T& v, Tally_impl& c) {
		std::map<value_type, std::size_t> result;
		for (const auto& i : v) {
			result[i]++;
		}
		return result;
	}
} Tally;

struct Reduce_impl {
	template <class T, class F> auto operator()(T memo, F f) {
		return Callable([memo, f](auto v) {
			auto acc = memo;
			for (auto i : v) {
				acc = f(acc, i);
			}
			return acc;
		});
	}
} Reduce;

struct Join_impl {
	auto operator()(std::string separater) {
		return Callable([&](auto v) {
			std::string result;
			bool first = true;
			for (const auto& i : v) {
				if (!std::exchange(first, false)) {
					result += separater;
				}
				result += std::to_string(i);
			}
			return result;
		});
	}
	template <class T> friend auto operator|(const T& v, Join_impl& c) {
		return v | c("");
	}
} Join;

struct At_impl {
	auto operator()(std::size_t l, std::size_t r) {
		return Callable(
		    [l, r](auto v) { return decltype(v)(std::begin(v) + l, std::begin(v) + r); });
	}
} At;
struct Slice_impl {
	auto operator()(std::size_t i, std::size_t cnt) {
		return Callable([i, cnt](auto v) {
			return decltype(v)(std::begin(v) + i, std::begin(v) + i + cnt);
		});
	}
} Slice;

struct Transpose_impl {
	template <class T>
	friend auto operator|(const std::vector<std::vector<T>>& v, Transpose_impl& c) {
		std::size_t h = v.size(), w = v.front().size();
		std::vector result(w, std::vector<T>(h));
		for (std::size_t i = 0; i < h; ++i) {
			assert(v[i].size() == w);
			for (std::size_t j = 0; j < w; ++j) {
				result[j][i] = v[i][j];
			}
		}
		return result;
	}
} Transpose;

template <class T> auto operator*(const std::vector<T>& a, std::size_t n) {
	T result;
	for (std::size_t i = 0; i < n; ++i) {
		result.insert(result.end(), a.begin(), a.end());
	}
	return result;
}
auto operator*(std::string a, std::size_t n) {
	std::string result;
	for (std::size_t i = 0; i < n; ++i) {
		result += a;
	}
	return result;
}

namespace internal {
	template <class T, class U, class = void> struct has_push_back : std::false_type {};
	template <class T, class U>
	struct has_push_back<T, U,
	                     std::void_t<decltype(std::declval<T>().push_back(std::declval<U>()))>>
	    : std::true_type {};
}  // namespace internal
template <
    class Container, class T,
    std::enable_if_t<internal::has_push_back<Container, T>::value, std::nullptr_t> = nullptr>
auto& operator<<(Container& continer, const T& val) {
	continer.push_back(val);
	return continer;
}
template <
    class Container, class T,
    std::enable_if_t<internal::has_push_back<Container, T>::value, std::nullptr_t> = nullptr>
auto operator+(Container continer, const T& val) {
	continer << val;
	return continer;
}
#line 4 "/home/yuruhiya/programming/library/Utility/functions.cpp"
#include <cmath>
#line 8 "/home/yuruhiya/programming/library/Utility/functions.cpp"

template <class T = long long> constexpr T TEN(std::size_t n) {
	T result = 1;
	for (std::size_t i = 0; i < n; ++i) result *= 10;
	return result;
}
template <
    class T, class U,
    std::enable_if_t<std::is_integral_v<T> && std::is_integral_v<U>, std::nullptr_t> = nullptr>
constexpr auto div_ceil(T n, U m) {
	return (n + m - 1) / m;
}
template <class T, class U> constexpr auto div_ceil2(T n, U m) {
	return div_ceil(n, m) * m;
}
template <class T> constexpr T triangle(T n) {
	return (n & 1) ? (n + 1) / 2 * n : n / 2 * (n + 1);
}
template <class T> constexpr T nC2(T n) {
	return (n & 1) ? (n - 1) / 2 * n : n / 2 * (n - 1);
}
template <class T, class U> constexpr auto middle(const T& l, const U& r) {
	return l + (r - l) / 2;
}
template <class T, class U, class V>
constexpr bool in_range(const T& v, const U& lower, const V& upper) {
	return lower <= v && v < upper;
}
template <class T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
constexpr bool is_square(T n) {
	T s = std::sqrt(n);
	return s * s == n || (s + 1) * (s + 1) == n;
}
template <class T = long long> constexpr T BIT(int b) {
	return T(1) << b;
}
template <class T> constexpr int BIT(T x, int i) {
	return (x & (T(1) << i)) ? 1 : 0;
}
template <class T> constexpr int Sgn(T x) {
	return (0 < x) - (0 > x);
}
template <class T> bool is_leap(T year) {
	return !(year % 4) && (year % 100 || !(year % 400));
}
template <class T, class U, std::enable_if_t<std::is_integral_v<U>, std::nullptr_t> = nullptr>
constexpr T Pow(T a, U n) {
	assert(n >= 0);
	T result = 1;
	while (n > 0) {
		if (n & 1) {
			result *= a;
			n--;
		} else {
			a *= a;
			n >>= 1;
		}
	}
	return result;
}
template <class T, class U, std::enable_if_t<std::is_integral_v<U>, std::nullptr_t> = nullptr>
constexpr T Powmod(T a, U n, T mod) {
	assert(n >= 0);
	if (a > mod) a %= mod;
	T result = 1;
	while (n > 0) {
		if (n & 1) {
			result = result * a % mod;
			n--;
		} else {
			a = a * a % mod;
			n >>= 1;
		}
	}
	return result;
}
template <class T> bool chmax(T& a, const T& b) {
	return a < b ? a = b, true : false;
}
template <class T> bool chmin(T& a, const T& b) {
	return a > b ? a = b, true : false;
}
template <class T> int sz(const T& v) {
	return v.size();
}
template <class T, class U> int lower_index(const T& a, const U& v) {
	return std::lower_bound(a.begin(), a.end(), v) - a.begin();
}
template <class T, class U, class F> int lower_index(const T& a, const U& v, const F& f) {
	return std::lower_bound(a.begin(), a.end(), v, f) - a.begin();
}
template <class T, class U> int upper_index(const T& a, const U& v) {
	return std::upper_bound(a.begin(), a.end(), v) - a.begin();
}
template <class T, class U, class F> int upper_index(const T& a, const U& v, const F& f) {
	return std::upper_bound(a.begin(), a.end(), v, f) - a.begin();
}
template <class T, class U = typename T::value_type> U Gcdv(const T& v) {
	return std::accumulate(std::next(v.begin()), v.end(), U(*v.begin()), std::gcd<U, U>);
}
template <class T, class U = typename T::value_type> U Lcmv(const T& v) {
	return std::accumulate(std::next(v.begin()), v.end(), U(*v.begin()), std::lcm<U, U>);
}
template <class T> T& Concat(T& v, const T& vec) {
	v.insert(v.end(), vec.begin(), vec.end());
	return v;
}
namespace internal {
	template <class T, std::size_t N>
	auto make_vector(std::vector<int>& sizes, const T& init) {
		if constexpr (N == 1) {
			return std::vector(sizes[0], init);
		} else {
			int size = sizes[N - 1];
			sizes.pop_back();
			return std::vector(size, make_vector<T, N - 1>(sizes, init));
		}
	}
}  // namespace internal
template <class T, std::size_t N>
auto make_vector(const int (&sizes)[N], const T& init = T()) {
	std::vector s(std::rbegin(sizes), std::rend(sizes));
	return internal::make_vector<T, N>(s, init);
}
template <class F> struct rec_lambda {
	F f;
	rec_lambda(F&& f_) : f(std::forward<F>(f_)) {}
	template <class... Args> auto operator()(Args&&... args) const {
		return f(*this, std::forward<Args>(args)...);
	}
};

namespace lambda {
	auto char_to_int = [](char c) {
		return c - '0';
	};
	auto lower_to_int = [](char c) {
		return c - 'a';
	};
	auto upper_to_int = [](char c) {
		return c - 'A';
	};
	auto int_to_char = [](int i) -> char {
		return '0' + i;
	};
	auto int_to_lower = [](int i) -> char {
		return 'a' + i;
	};
	auto int_to_upper = [](int i) -> char {
		return 'A' + i;
	};
	auto is_odd = [](auto n) {
		return n % 2 == 1;
	};
	auto is_even = [](auto n) {
		return n % 2 == 0;
	};
	auto is_positive = [](auto n) {
		return n > 0;
	};
	auto is_negative = [](auto n) {
		return n < 0;
	};
	auto increment = [](auto n) {
		return ++n;
	};
	auto decrement = [](auto n) {
		return --n;
	};
	auto self = [](const auto& n) {
		return n;
	};
	auto first = [](const auto& n) {
		return n.first;
	};
	auto second = [](const auto& n) {
		return n.second;
	};
	template <class T> auto cast() {
		return [](const auto& n) {
			return static_cast<T>(n);
		};
	};
	template <class T> auto equal_to(const T& x) {
		return [x](auto y) {
			return x == y;
		};
	}
	template <std::size_t I> auto get() {
		return [](const auto& n) {
			return std::get<I>(n);
		};
	}
	template <class F> auto cmp(F&& f) {
		return [f](const auto& a, const auto& b) {
			return f(a) < f(b);
		};
	}
}  // namespace lambda
#line 8 "/home/yuruhiya/programming/library/template.cpp"
#if __has_include(<library/dump.hpp>)
#include <library/dump.hpp>
#define LOCAL
#else
#define dump(...) ((void)0)
#define dump2(...) ((void)0)
#endif
#line 2 "/home/yuruhiya/programming/library/Utility/oj_local.cpp"
template <class T> constexpr T oj_local(const T& oj, const T& local) {
#ifndef LOCAL
	return oj;
#else
	return local;
#endif
}
#line 16 "/home/yuruhiya/programming/library/template.cpp"
#include <bits/stdc++.h>
#line 6 "/home/yuruhiya/programming/library/Graph/GraphTemplate.cpp"

using Weight = long long;
constexpr Weight INF = std::numeric_limits<Weight>::max();
struct Edge {
	int to;
	Weight cost;
	Edge() : to(-1), cost(-1) {}
	Edge(int _to, Weight _cost = 1) : to(_to), cost(_cost) {}
	friend bool operator<(const Edge& e1, const Edge& e2) {
		return e1.cost < e2.cost;
	}
	friend bool operator>(const Edge& e1, const Edge& e2) {
		return e1.cost > e2.cost;
	}
	friend std::ostream& operator<<(std::ostream& os, const Edge& e) {
		return os << "->" << e.to << '(' << e.cost << ')';
	}
};
using UnWeightedGraph = std::vector<std::vector<int>>;
using Graph = std::vector<std::vector<Edge>>;
struct Edge2 {
	int from, to;
	Weight cost;
	Edge2() : from(-1), to(-1), cost(0) {}
	Edge2(int _from, int _to, Weight _cost) : from(_from), to(_to), cost(_cost) {}
	friend bool operator<(const Edge2& e1, const Edge2& e2) {
		return e1.cost < e2.cost;
	}
	friend bool operator>(const Edge2& e1, const Edge2& e2) {
		return e1.cost > e2.cost;
	}
	friend std::ostream& operator<<(std::ostream& os, const Edge2& e) {
		return os << e.from << "->" << e.to << '(' << e.cost << ')';
	}
};
using UnWeightedEdges = std::vector<std::pair<int, int>>;
using Edges = std::vector<Edge2>;
using Matrix = std::vector<std::vector<Weight>>;

auto add_edge(UnWeightedGraph& graph, int v, int u) {
	graph[v].push_back(u);
	graph[u].push_back(v);
}
auto add_edge(Graph& graph, int v, int u, Weight cost) {
	graph[v].emplace_back(u, cost);
	graph[u].emplace_back(v, cost);
}
auto to_graph(const UnWeightedGraph& graph, Weight cost = 1) {
	Graph result(graph.size());
	for (std::size_t i = 0; i < graph.size(); ++i) {
		for (int v : graph[i]) {
			result[i].emplace_back(v, cost);
		}
	}
	return result;
}
auto to_unweighted_graph(const Graph& graph) {
	UnWeightedGraph result(graph.size());
	for (std::size_t i = 0; i < graph.size(); ++i) {
		for (auto [v, cost] : graph[i]) {
			result[i].push_back(v);
		}
	}
	return result;
}
auto to_edges(const UnWeightedGraph& graph, bool unique = false) {
	std::vector<std::pair<int, int>> edges;
	for (std::size_t i = 0; i < graph.size(); ++i) {
		for (int v : graph[i]) {
			if (!unique || static_cast<int>(i) < v) edges.emplace_back(i, v);
		}
	}
	return edges;
}
auto to_edges(const Graph& graph) {
	Edges edges;
	for (std::size_t i = 0; i < graph.size(); ++i) {
		for (auto [v, cost] : graph[i]) {
			edges.emplace_back(i, v, cost);
		}
	}
	return edges;
}
#line 5 "/home/yuruhiya/programming/library/DP/Knapsack01.cpp"

template <class T>
std::vector<T> Knapsack01(int n, int weight_limit, const std::vector<T>& value,
                          const std::vector<int>& weight) {
	assert(n == static_cast<int>(value.size()));
	assert(n == static_cast<int>(weight.size()));
	std::vector<T> dp(weight_limit + 1, 0);
	for (int i = 0; i < n; ++i) {
		for (int j = weight_limit; j >= 0; --j) {
			if (j - weight[i] >= 0) dp[j] = std::max(dp[j], dp[j - weight[i]] + value[i]);
		}
	}
	return dp;
}
#line 4 "a.cpp"
using namespace std;

int main() {
	int n = in, k = in;
	Graph g(n);
	rep(i, n - 1) {
		int a = in--, b = in--, c = in;
		g[a].emplace_back(b, c);
		g[b].emplace_back(a, c);
	}

	VI value, weight;
	auto dfs = [&](auto&& f, int v, int p) -> int {
		bool is_leaf = true;
		int leaf_cnt = 0;
		for (auto e : g[v]) {
			if (e.to == p) continue;
			is_leaf = false;

			int res = f(f, e.to, v);
			weight << e.cost;
			value << e.cost * res;

			leaf_cnt += res;
		}
		return leaf_cnt + is_leaf;
	};
	dfs(dfs, 0, -1);
	dump(value, weight);
	out((value | Sum) + (Knapsack01(n - 1, k, value, weight) | Max));
}