/*------------------------------------------------------------


		   Welcome to my program!
		   DON'T HACK PLEASE!!!!!!!!

		　　∧＿∧        AtCoder / Codeforces / yukicoder
		　 ( 　･ω･)
		＿(__つ/￣￣￣ /
		　　＼/　　　　 /  C++ GCC 12.3.0 + Boost 1.82.0
		　　　　￣￣￣￣￣
				   Let's write Code!


------------------------------------------------------------*/

// Return Code 139(out_of_range)が出たら試す
// #define _GLIBCXX_DEBUG

// Boost系
// #include <boost/multiprecision/cpp_int.hpp> 			 // クソでか整数
// #include <boost/multiprecision/cpp_dec_float.hpp>	 // クソでか小数
// #include <boost/math/constants/constants.hpp>		 // ウルトラ円周率
// #include <boost/date_time/gregorian/gregorian.hpp>	 // 申し訳程度の日付演算

// #include <boost/algorithm/string/split.hpp>			 // 文字列をcharで分割(split)
// #include <boost/algorithm/string/classification.hpp> // 上の文字指定に必要
// #include <boost/algorithm/string/iter_find.hpp>		 // 文字列をstringで分割(iter_split)
// #include <boost/algorithm/string/finder.hpp>		 // 上の文字列指定に必要

// #include <boost/algorithm/string/join.hpp>			 // 他の言語のjoinを実装するやつ
// #include <boost/algorithm/string/replace.hpp>		 // 全部置き換え

// #include <boost/dynamic_bitset.hpp>					 // サイズ変更可bitset
// #include <boost/range/algorithm/for_each.hpp>		 // 関数でforeach文

// #include <boost/assert.hpp> // 上位デバッグ

#pragma region AtCoder Template

#include <bits/stdc++.h>
using namespace std;

// ローカル環境チェック
#if __has_include("./lib/debug_print.hpp")
#define LOCAL 1
#endif

// AC Library(C++17では使えないので注意)
#if defined(LOCAL) || defined(ATCODER)
#include <atcoder/all>
using namespace atcoder;
#endif

#ifdef LOCAL
#include "./lib/debug_print.hpp"
#define debug(...) debug_print::multi_print(#__VA_ARGS__, __VA_ARGS__)
#else
#define debug(...) (static_cast<void>(0))
#endif

#ifdef BOOST_MP_CPP_INT_HPP
namespace mu = boost::multiprecision;
#endif

// 高速化
#pragma GCC target("avx,avx2")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")

#define fastio                         \
	cin.tie(nullptr);                  \
	ios::sync_with_stdio(false);       \
	cout << fixed << setprecision(15); \
	srand((unsigned)time(NULL));

// 型省略
using uint = unsigned;
using ll = long long;
// using ll = __int128_t;
using ull = unsigned long long;
using ld = long double;
using pll = pair<ll, ll>;
using vll = vector<ll>;
using vb = vector<bool>;
using vc = vector<char>;
using vs = vector<string>;
using vd = vector<double>;
using vld = vector<ld>;
using vull = vector<ull>;
using vpll = vector<pll>;
using pdd = pair<ld, ld>;
using psl = pair<string, ll>;
using pcl = pair<char, ll>;
using vvll = vector<vll>;
using vvc = vector<vc>;
using vvs = vector<vs>;
using vvb = vector<vb>;
using vvld = vector<vld>;
using vvd = vector<vd>;
using mll = map<ll, ll>;
using mcl = map<char, ll>;
using msl = map<string, ll>;
using sll = set<ll>;
using spll = set<pair<ll, ll>>;
using spdd = set<pair<ld, ld>>;
using stll = stack<ll>;
using qll = queue<ll>;
using qd = queue<ld>;
using qs = queue<string>;
using qc = queue<char>;
using int128_t = __int128_t;

template <typename Tp1, typename Tp2>
using unmap = unordered_map<Tp1, Tp2>;

template <typename Tp>
using unset = unordered_set<Tp>;

template <typename Tp>
using reverse_queue = priority_queue<Tp, vector<Tp>, greater<Tp>>;

// マクロ
#define rep(i, n) for (ll i = 0; i < (ll)(n); i++)
#define rrep(i, n) for (ll i = (n) - 1; i >= 0; i--)
#define irep(i, n) for (ll i = 1; i <= (ll)(n); i++)
#define arep(i, a, n) for (ll i = (a); i < (ll)(n); i++)
#define adrep(i, a, d, n) for (ll i = (a); i < (ll)(n); i += d)
#define until(b) while (!(b))

// 省略define
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define fi first
#define se second
#define endl "\n"
#define br break
#define el else
#define elif else if

template <typename T>
inline void YESNO(T b)
{
	cout << (b ? "YES" : "NO") << endl;
}

template <typename T>
inline void yesno(T b)
{
	cout << (b ? "yes" : "no") << endl;
}

template <typename T>
inline void YesNo(T b)
{
	cout << (b ? "Yes" : "No") << endl;
}

template <typename T, typename tr, typename fal>
inline void outif(T b, tr tru, fal fals)
{
	if (b)
	{
		cout << tru << endl;
	}
	else
	{
		cout << fals << endl;
	}
}

#define exit exit(0)
#define co(x) cout << (x) << endl

// 定数
const string sl = "";
const char cl = '\0';
const ll nl = 0LL;
const ll INFINT = 2047483647;
const ll INFLL = 1023372036854775807LL; // だいたい
const ll mod1 = pow(10, 9) + 1;
const char sp = ' ';
const ll j2_32 = pow(2, 32);
const ll j2_m32 = pow(2, -32);
const ll j2_10 = pow(2, 10);
const vector<int> dx = {0, 0, 1, -1};
const vector<int> dy = {1, -1, 0, 0};
const vector<int> ex = {-1, -1, -1, 0, 0, 1, 1, 1};
const vector<int> ey = {-1, 0, 1, -1, 1, -1, 0, 1};
const string spa = " ";
const bool except = true;

// 色々なテンプレ(完全コピペ)

template <class T>
size_t HashCombine(const size_t seed, const T &v)
{
	return seed ^ (std::hash<T>()(v) + 0x9e3779b9 + (seed << 6) + (seed >> 2));
}
/* pair用 */
template <class T, class S>
struct std::hash<std::pair<T, S>>
{
	size_t operator()(const std::pair<T, S> &keyval) const noexcept
	{
		return HashCombine(std::hash<T>()(keyval.first), keyval.second);
	}
};
/* vector用 */
template <class T>
struct std::hash<std::vector<T>>
{
	size_t operator()(const std::vector<T> &keyval) const noexcept
	{
		size_t s = 0;
		for (auto &&v : keyval)
			s = HashCombine(s, v);
		return s;
	}
};
/* tuple用 */
template <int N>
struct HashTupleCore
{
	template <class Tuple>
	size_t operator()(const Tuple &keyval) const noexcept
	{
		size_t s = HashTupleCore<N - 1>()(keyval);
		return HashCombine(s, std::get<N - 1>(keyval));
	}
};
template <>
struct HashTupleCore<0>
{
	template <class Tuple>
	size_t operator()(const Tuple &keyval) const noexcept { return 0; }
};
template <class... Args>
struct std::hash<std::tuple<Args...>>
{
	size_t operator()(const tuple<Args...> &keyval) const noexcept
	{
		return HashTupleCore<tuple_size<tuple<Args...>>::value>()(keyval);
	}
};

std::string
operator""_s(char const *str, std::size_t)
{
	return str;
}

std::string
operator*(std::string const &str, int n)
{
	if (n < 1)
		return "";
	std::string result;
	result.reserve(str.length() * n);
	for (int i = 0; i < n; ++i)
	{
		result += str;
	}
	return result;
}

// https://kenkoooo.hatenablog.com/entry/2016/11/30/163533
std::ostream &operator<<(std::ostream &dest, __int128_t value)
{
	std::ostream::sentry s(dest);
	if (s)
	{
		__uint128_t tmp = value < 0 ? -value : value;
		char buffer[128];
		char *d = std::end(buffer);
		do
		{
			--d;
			*d = "0123456789"[tmp % 10];
			tmp /= 10;
		} while (tmp != 0);
		if (value < 0)
		{
			--d;
			*d = '-';
		}
		int len = std::end(buffer) - d;
		if (dest.rdbuf()->sputn(d, len) != len)
		{
			dest.setstate(std::ios_base::badbit);
		}
	}
	return dest;
}

__int128 parse(string &s)
{
	__int128 ret = 0;
	for (ull i = 0; i < s.length(); i++)
		if ('0' <= s[i] && s[i] <= '9')
			ret = 10 * ret + s[i] - '0';

	if (s[0] == '-')
	{
		ret = -ret;
	}

	return ret;
}

istream &operator>>(std::istream &is, __int128_t &value)
{
	string tmp;
	is >> tmp;

	value = parse(tmp);

	return is;
}

// 関数類

/**
 * @brief 素数をチェックします
 *
 * @param num 判定する数値
 * @return bool 素数かどうか
 */
inline bool isprime(const ull num) noexcept(except)
{
	if (num < 2)
		return false;
	else if (num == 2)
		return true;
	else if (num % 2 == 0)
		return false;

	double sqrtNum = sqrt(num);
	for (int i = 3; i <= sqrtNum; i += 2)
	{
		if (num % i == 0)
		{
			return false;
		}
	}
	return true;
}

/**
 * @brief char型の数値をint型に変換します
 *
 * @param c 変換する文字
 * @return int 変換した数値
 */
inline int ctoi(const char c) noexcept(except)
{
	if (c >= '0' && c <= '9')
		return c - '0';
	return 0;
}

/**
 * @brief 1+2+3+4+...n
 *
 * @param n
 * @return int
 */
inline ull minisum(const ull n) noexcept(except)
{
	return n * (n + 1ULL) / 2ULL;
}

/**
 * @brief 数値を桁数で0埋めします
 *
 * @tparam T 桁数の型
 * @param i 桁数
 * @param s 埋める文字列
 * @return string 0埋め後の文字列
 */
inline string zerou(const ull i, string s) noexcept(except)
{
	while (s.size() != i)
		s = '0' + s;
	return s;
}

/**
 * @brief T型をchar型に変換します
 *
 * @tparam T 変換する型
 * @param i 変換する数値
 * @return char 変換後の文字
 */
inline char to_char(const ull i) noexcept(except)
{
	assert(i <= 9);
	return '0' + i;
}

/**
 * @brief i < j の場合iをjで置き換えます
 *
 * @tparam T1_ iの型
 * @tparam T2_ jの型
 * @param i 上書きする変数
 * @param j 比較する変数
 * @return bool 置き換えたかどうか
 */
template <typename T1_, typename T2_>
inline bool chmax(T1_ &i, const T2_ j) noexcept(except)
{
	if (i < j)
	{
		i = j;
		return true;
	}
	return false;
}

/**
 * @brief i > j の場合iをjで置き換えます
 *
 * @tparam T1_ iの型
 * @tparam T2_ jの型
 * @param i 上書きする変数
 * @param j 比較する変数
 * @return bool 置き換えたかどうか
 */
template <typename T1_, typename T2_>
inline bool chmin(T1_ &i, const T2_ j) noexcept(except)
{
	if (i > j)
	{
		i = j;
		return true;
	}
	return false;
}

/**
 * @brief 配列内の全要素を出力します
 *
 * @tparam T 配列の型(vector<T>)
 * @param v 配列
 * @param s 区切り文字(規定ではスペース)
 * @author https://zenn.dev/antyuntyun
 */
template <typename T>
inline void print(const vector<T> &v, const string &s = " ") noexcept(except)
{
	rep(i, v.size()) cout << v[i] << (i != (ll)v.size() - 1 ? s : "\n");
}

template <typename A, typename B>
inline void print(const vector<pair<A, B>> &v, const string &s = "\n") noexcept(except)
{
	rep(i, v.size()) cout << v[i].first << " " << v[i].second << s;
}

/// @brief 二次元配列の全要素を出力します
/// @tparam T 配列の型(vector<vector<T>>)
/// @param v 二次元配列
/// @param s 区切り文字
template <typename T>
inline void print(const vector<vector<T>> &v, string const &s = " ") noexcept(except)
{
	rep(i, v.size())
	{
		rep(j, v[i].size()) cout << v[i][j] << (j != (ll)v[i].size() - 1 ? s : "\n");
	}
}

template <typename T>
inline istream &operator>>(istream &os, vector<T> &v)
{
	assert(v.size() != 0);
	rep(i, v.size())
	{
		cin >> v[i];
	}

	return os;
}

/**
 * @brief 文字列を反転した文字列を返します
 *
 * @param s 反転する文字列
 * @return string 反転後の文字列
 */
inline string srev(string s) noexcept(except)
{
	reverse(all(s));
	return s;
}

/// @brief long longでべき乗します
/// @param x x^nのx
/// @param n x^nのn
/// @return x^n
inline unsigned long long pow_ll(unsigned long long x, unsigned long long n) noexcept(except)
{
	ull ret = 1;
	while (n > 0)
	{
		if (n & 1)
			ret *= x;
		x *= x;
		n >>= 1;
	}

	return ret;
}

/// @brief N進数の文字から10進数の数値に変換します
/// @param c N進数の文字
/// @return 変換した10進数の数値
inline int ntodec(const char c)
{
	switch (c)
	{
	case '0':
		return 0;
	case '1':
		return 1;
	case '2':
		return 2;
	case '3':
		return 3;
	case '4':
		return 4;
	case '5':
		return 5;
	case '6':
		return 6;
	case '7':
		return 7;
	case '8':
		return 8;
	case '9':
		return 9;
	case 'A':
		return 10;
	case 'B':
		return 11;
	case 'C':
		return 12;
	case 'D':
		return 13;
	case 'E':
		return 14;
	case 'F':
		return 15;
	case 'G':
		return 16;
	case 'H':
		return 17;
	case 'I':
		return 18;
	case 'J':
		return 19;
	case 'K':
		return 20;
	case 'L':
		return 21;
	case 'M':
		return 22;
	case 'N':
		return 23;
	case 'O':
		return 24;
	case 'P':
		return 25;
	case 'Q':
		return 26;
	case 'R':
		return 27;
	case 'S':
		return 28;
	case 'T':
		return 29;
	case 'U':
		return 30;
	case 'V':
		return 31;
	case 'W':
		return 32;
	case 'X':
		return 33;
	case 'Y':
		return 34;
	case 'Z':
		return 35;
	default:
		return -1;
	}
}

/// @brief 10進数の数値をN進数の文字に変換します
/// @param n 10進数の数値
/// @return N進数の文字
inline char decton(const int n)
{
	switch (n)
	{
	case 0:
		return '0';
	case 1:
		return '1';
	case 2:
		return '2';
	case 3:
		return '3';
	case 4:
		return '4';
	case 5:
		return '5';
	case 6:
		return '6';
	case 7:
		return '7';
	case 8:
		return '8';
	case 9:
		return '9';
	case 10:
		return 'A';
	case 11:
		return 'B';
	case 12:
		return 'C';
	case 13:
		return 'D';
	case 14:
		return 'E';
	case 15:
		return 'F';
	case 16:
		return 'G';
	case 17:
		return 'H';
	case 18:
		return 'I';
	case 19:
		return 'J';
	case 20:
		return 'K';
	case 21:
		return 'L';
	case 22:
		return 'M';
	case 23:
		return 'N';
	case 24:
		return 'O';
	case 25:
		return 'P';
	case 26:
		return 'Q';
	case 27:
		return 'R';
	case 28:
		return 'S';
	case 29:
		return 'T';
	case 30:
		return 'U';
	case 31:
		return 'V';
	case 32:
		return 'W';
	case 33:
		return 'X';
	case 34:
		return 'W';
	case 35:
		return 'Z';
	default:
		return '\0';
	}
}

/// @brief N進数の文字列をM進数に直して出力します。
/// @param str N進数の文字
/// @param n 文字の進数
/// @param m 出力の進数
/// @return M進数の文字
inline string n_ary(const string &str, const int n, const int m)
{
	unsigned long tmp = 0;
	string ret;

	for (unsigned long long i = 0; i < str.length(); i++)
	{
		tmp += (unsigned long)ntodec(str[str.length() - 1 - i]) * pow_ll(n, i);
	}

	if (tmp == 0)
		return "0";
	while (tmp != 0)
	{
		ret = decton(tmp % m) + ret;
		tmp /= m;
	}
	return ret;
}

/// @brief
/// @tparam T nの型
/// @param n 素因数分解する数
/// @return 不明
template <typename T>
inline map<T, T> __prime_factor(T n)
{
	map<T, T> ret;
	for (T i = 2; i * i <= n; i++)
	{
		T tmp = 0;
		while (n % i == 0)
		{
			tmp++;
			n /= i;
		}
		ret[i] = tmp;
	}
	if (n != 1)
		ret[n] = 1;
	return ret;
}

// O(sqrt(N))
vector<pair<long long, long long>> prime_factor(long long N)
{
	// 答えを表す可変長配列
	vector<pair<long long, long long>> res;

	// √N まで試し割っていく
	for (long long p = 2; p * p <= N; ++p)
	{
		// N が p で割り切れないならばスキップ
		if (N % p != 0)
		{
			continue;
		}

		// N の素因数 p に対する指数を求める
		int e = 0;
		while (N % p == 0)
		{
			// 指数を 1 増やす
			++e;

			// N を p で割る
			N /= p;
		}

		// 答えに追加
		res.emplace_back(p, e);
	}

	// 素数が最後に残ることがありうる
	if (N != 1)
	{
		res.emplace_back(N, 1);
	}
	return res;
}

/// @brief Nの約数の数を求めます
/// @tparam T Nの型
/// @param N 約数の数を求める数
/// @return Nの約数の数
template <typename T>
inline T divisor_num(const T N)
{
	map<T, T> pf = __prime_factor(N);
	T ret = 1;
	for (auto p : pf)
	{
		ret *= (p.second + 1);
	}
	return ret;
}

/// @brief nの約数を全列挙します。(計算量: O(sqrt(N)))
/// @param n 全列挙する約数
/// @return nの約数
inline vll divisor(const ll n)
{
	vll ret;
	for (ll i = 1; i * i <= n; i++)
	{
		if (n % i == 0)
		{
			ret.push_back(i);
			if (i * i != n)
				ret.push_back(n / i);
		}
	}
	sort(ret.begin(), ret.end());
	return ret;
}

/// @brief 文字列が数字のみか判定します O(|S|)
/// @param s 判定する文字列
/// @return 数値でできた文字列かどうか
inline bool isint(const string &s) noexcept(except)
{
	rep(i, s.size())
	{
		if (!isdigit(s[i]))
		{
			return false;
		}
	}

	return true;
}

/// @brief 二次元配列を90度時計回りに回転する
/// @tparam T 配列の型(vector<vector<T>>)
/// @param arr 二次元配列
/// @return 返り値
template <typename T>
inline vector<vector<T>> rot90(const vector<vector<T>> &A)
{
	ll N = A.size(), M = A[0].size();
	vector<vector<T>> ret(M, vector<T>(N));

	ll _i = 0, _j = 0;

	rep(j, M)
	{
		for (ll i = N - 1; i >= 0; i--)
		{
			ret[_i][_j] = A[i][j];
			_j++;
		}
		_j = 0;
		_i++;
	}

	return ret;
}

/// @brief 回文かどうか判定
/// @param str 文字列
/// @return 回文かどうか
inline bool ispalind(const string &str) noexcept(except)
{
	int n = str.length();
	for (int i = 0; i < n / 2; i++)
	{
		if (str[i] != str[n - i - 1])
		{
			return false;
		}
	}
	return true;
}

/// @brief startからnまでの順列を生成
/// @param n 最大値
/// @param start 開始値
/// @return startからnまでの順列
inline vector<ll> range(const ll n, const ll start = 0)
{
	vector<ll> ret(n - start);
	ll oi = 0;
	for (ll i = start; i <= n; i++)
	{
		ret[oi] = i;
		oi++;
	}

	return ret;
}

/// @brief 10進法で表した時の各桁の和を求めます
/// @param s 10進法で表した文字列
/// @return 各桁の和
inline ll csum(const string &s) noexcept(except)
{
	ll ret = 0;
	rep(i, s.size())
	{
		ret += ctoi(s[i]);
	}

	return ret;
}

/// @brief csumの数値用の補完関数
/// @param n 数値
/// @return 各桁の和
inline ll csum(const ll n) noexcept(except)
{
	return csum(to_string(n));
}

/// @brief 階乗を計算する
/// @param n nの階乗
/// @return nの階乗
inline ll fact(const ll n) noexcept(except)
{
	ll ret = 1;
	rep(i, n)
	{
		ret *= i + 1;
	}
	return ret;
}

/// @brief 平方数かどうかを判定
/// @param N 判定する数
/// @return 平方数かどうか
inline bool is_squere(const ll N) noexcept(except)
{
	long long r = (long long)floor(sqrt((long double)N)); // 切り捨てした平方根
	return (r * r) == N;
}

/// @brief 一次元の累積和を返します
/// @tparam T vectorの型
/// @param v 加工する前の配列
/// @return 加工後の配列(長さは |v|+1 となります。)
template <typename T>
inline vector<T> cumulative(const vector<T> &v) noexcept(except)
{
	vector<T> cum(v.size() + 1);
	cum[0] = 0;
	for (int i = 1; i <= v.size(); i++)
	{
		cum[i] = cum[i - 1] + v[i - 1];
	}
	return cum;
}

/// @brief 二次元の累積和を返します
/// @tparam T vector<vector<>>の型
/// @param v 加工前の配列
/// @return 加工後の配列(長さはそれぞれ+1になります)
template <typename T>
inline vector<vector<T>> cumulative(const vector<vector<T>> &v)
{
	assert(v.size() > 0);
	ll H = v.size(), W = v[0].size();
	vector<vector<T>> ret(H + 1, vector<T>(W + 1, 0));
	for (int i = 1; i <= H; i++)
	{
		for (int j = 1; j <= W; j++)
		{
			ret[i][j] = ret[i][j - 1] + v[i - 1][j - 1];
		}
	}

	for (int j = 1; j <= W; j++)
	{
		for (int i = 1; i <= H; i++)
		{
			ret[i][j] += ret[i - 1][j];
		}
	}

	return ret;
}

/// @brief ランダムな数値を返す
/// @param l 最小値
/// @param r 最大値
/// @return
inline ll randint(const ll l, const ll r) noexcept(except)
{
	if (l == r)
		return l;
	return l + (rand() % (r - l));
}

/// @brief ランダムな小数を返す(0<=x<=1)
/// @return 0<=x<=1
inline ld randd() noexcept(except)
{
	return 1.0L * rand() / RAND_MAX;
}

/// @brief 高速全探索 O(log N)
/// @tparam _Tp 配列の型
/// @param v 配列
/// @param x 探索するやつ
/// @return 数
template <typename _Tp>
inline long long bound_count(const vector<_Tp> &v, const _Tp &x) noexcept(except)
{
	auto l = lower_bound(v.begin(), v.end(), x);
	auto u = upper_bound(v.begin(), v.end(), x);

	if (*l != x)
	{
		return 0;
	}

	if (u == v.end())
	{
		return v.size() - (l - v.begin());
	}
	else
	{
		return (u - v.begin()) - (l - v.begin());
	}
}

/// @brief 配列の最近値を求める
/// @tparam T 配列の型
/// @param v 配列
/// @param x 最近値を求める値
/// @return xの最近値
template <typename T>
inline T recent(const vector<T> &v, const T &x)
{
	auto it = lower_bound(all(v), x);

	if (it == v.end())
		return *prev(v.end(), 1);
	else
	{
		if (it == v.begin())
			return *v.begin();
		else
		{
			if (abs(*it - x) < abs(*prev(it, 1) - x))
				return *it;
			else
				return *prev(it, 1);
		}
	}
}

template <typename T>
inline vector<vector<T>> make_vec2(const ull H, const ull W, const T &init)
{
	return vector<vector<T>>(H, vector<T>(W, init));
}

template <typename T>
inline vector<vector<T>> make_vec2(const ull H, const ull W)
{
	return vector<vector<T>>(H, vector<T>(W));
}

/// @brief 文字列圧縮
/// @param str 圧縮する文字列
/// @return 圧縮後
inline vector<pair<char, ull>> rlencode(const string &str) noexcept(except)
{
	ull n = (ull)str.size();
	vector<pair<char, ull>> ret;
	for (ull l = 0; l < n;)
	{
		ull r = l + 1;
		for (; r < n && str[l] == str[r]; r++)
		{
		};
		ret.push_back({str[l], r - l});
		l = r;
	}
	return ret;
}

template <typename T>
inline map<T, ll> counter(const vector<T> &v) noexcept(except)
{
	map<T, ll> dat;
	rep(i, v.size())
	{
		dat[v[i]]++;
	}

	return dat;
}

inline map<char, ll> counter(const string &s) noexcept(except)
{
	map<char, ll> dat;
	rep(i, s.size())
	{
		dat[s[i]]++;
	}

	return dat;
}

/// @brief ユークリッド距離
/// @param x1
/// @param y1
/// @param x2
/// @param y2
/// @return
inline ld euclidean(const ld x1, const ld y1, const ld x2, const ld y2) noexcept(except)
{
	ld dx = x2 - x1;
	ld dy = y2 - y1;

	ld distance = sqrt(dx * dx + dy * dy);

	return distance;
}

/// @brief 配列の範囲(閉区間)に属する値の個数を計算
/// @tparam T 配列の値型
/// @param v 配列
/// @param l 左端
/// @param r 右端
/// @return
template <typename T>
inline ll lencnt(const vector<T> &v, const T &l, const T &r)
{
	return upper_bound(all(v), r) - lower_bound(all(v), l);
}

using GraphKey = ll;

struct CostEdge
{
	GraphKey to;
	ll cost;
};

ostream &operator<<(ostream &os, const CostEdge &cost)
{
	os << "{ to: " << cost.to << ", cost: " << cost.cost << " }";

	return os;
}

using Edge = GraphKey;

using Graph = vector<vector<Edge>>;
using CostGraph = vector<vector<CostEdge>>;

inline CostGraph convert_costgraph(const Graph &g) noexcept
{
	CostGraph ans(g.size());
	rep(i, g.size())
	{
		rep(j, g[i].size())
		{
			ans[i].emplace_back(CostEdge{g[i][j], 1});
		}
	}

	return ans;
}

inline pair<GraphKey, ll> tree_diamiter_dfs(const CostGraph &G, ll u, ll par)
{ // 最遠点間距離と最遠点を求める
	pair<GraphKey, ll> ret = make_pair((GraphKey)0, u);
	for (auto e : G[u])
	{
		if (e.to == par)
			continue;
		auto next = tree_diamiter_dfs(G, e.to, u);
		next.first += e.cost;
		ret = max(ret, next);
	}
	return ret;
}

inline GraphKey tree_diamiter(const CostGraph &G)
{
	pair<GraphKey, ll> p = tree_diamiter_dfs(G, 0LL, -1LL);
	pair<GraphKey, ll> q = tree_diamiter_dfs(G, p.second, -1LL);
	return q.first;
}

inline GraphKey tree_diamiter(const Graph &G)
{
	return tree_diamiter(convert_costgraph(G));
}

inline CostEdge make_cost(const GraphKey to, const ll cost) noexcept
{
	return CostEdge{to, cost};
}

inline vector<ll> dijkstra(const CostGraph &G, ll start = 0, ll init = 0)
{
	ll n = G.size();
	assert(0 <= start && start < n);
	vector<bool> kakutei(n, false);
	vll cur(n, INFLL);

	reverse_queue<pll> q;
	cur[start] = init;
	q.push(make_pair(cur[start], start));

	while (!q.empty())
	{
		ll pos = q.top().second;
		q.pop();

		if (kakutei[pos])
			continue;

		kakutei[pos] = true;
		rep(i, G[pos].size())
		{
			ll nex = G[pos][i].to;
			ll cost = G[pos][i].cost;
			if (cur[nex] > cur[pos] + cost)
			{
				cur[nex] = cur[pos] + cost;
				q.push(make_pair(cur[nex], nex));
			}
		}
	}

	return cur;
}

inline vector<ll> dijkstra(const CostGraph &G, vll &prv, ll start = 0, ll init = 0)
{
	ll n = G.size();
	assert(0 <= start && start < n);
	vector<bool> kakutei(n, false);
	vll cur(n, INFLL);
	prv.resize(G.size(), -1);

	reverse_queue<pll> q;
	cur[start] = init;
	q.push(make_pair(cur[start], start));

	while (!q.empty())
	{
		ll pos = q.top().second;
		q.pop();

		if (kakutei[pos])
			continue;

		kakutei[pos] = true;
		rep(i, G[pos].size())
		{
			ll nex = G[pos][i].to;
			ll cost = G[pos][i].cost;
			if (cur[nex] > cur[pos] + cost)
			{
				cur[nex] = cur[pos] + cost;
				prv[nex] = pos;
				q.push(make_pair(cur[nex], nex));
			}
		}
	}

	return cur;
}

inline vector<ll> get_path(const vector<ll> &prev, ll t)
{
	vector<ll> path;
	for (ll cur = t; cur != -1; cur = prev[cur])
	{
		path.push_back(cur);
	}
	reverse(path.begin(), path.end()); // 逆順なのでひっくり返す
	return path;
}

inline vector<ll> dijkstra(const Graph &G, ll start = 0, ll init = 0)
{
	return dijkstra(convert_costgraph(G), start, init);
}

inline vector<vector<ll>> warshall_floyd(const CostGraph &G)
{
	ll n = G.size();
	vvll d = make_vec2<ll>(n, n, INFLL);

	rep(i, n)
	{
		d[i][i] = 0;
	}

	rep(i, n)
	{
		rep(j, G[i].size())
		{
			d[i][G[i][j].to] = G[i][j].cost;
		}
	}

	rep(k, n)
	{
		rep(i, n)
		{
			rep(j, n)
			{
				d[i][j] = min(d[i][j], d[i][k] + d[k][j]);
			}
		}
	}

	return d;
}

inline vector<vector<ll>> warshall_floyd(const Graph &G)
{
	return warshall_floyd(convert_costgraph(G));
}

template <ull bit, ull n>
class CustomBit
{
public:
	CustomBit(ull val = 0)
	{
		this->__val = val;
		this->__max_val = pow_ll(bit, n) - 1;
		this->__reload();
	}

	ull to_ull() const
	{
		return this->__val;
	}

	// O(1)
	ull max_val() const
	{
		return this->__max_val;
	}

	// O(1)
	array<ull, n> get_all() const
	{
		return this->__dat;
	}

	// O(1)
	ull get(ull x) const
	{
		assert(x < n);

		return this->__dat[x];
	}

	// O(1)
	constexpr ull size() const
	{
		return n;
	}

	constexpr void set(ull x, ull val)
	{
		assert(val < bit);
		this->__dat[x] = val;

		this->__reload_val();
	}

	CustomBit &operator++(int)
	{
		this->__val++;
		this->__reload();

		return *this;
	}

	CustomBit &operator++()
	{
		auto tmp = *this;
		++this->__val;

		this->__reload();

		return tmp;
	}

private:
	ull __val;
	array<ull, n> __dat;
	ull __max_val;

	void __reload()
	{
		assert(0 <= this->__val && this->__val <= this->__max_val);
		auto tmp = this->__val;
		for (ll i = 0; i < n; ++i)
		{
			this->__dat[i] = tmp % bit;
			tmp /= bit;
		}
	}

	void __reload_val()
	{
		this->__val = 0;
		ull a = 1;
		for (ll i = 0; i < n; ++i)
		{
			this->__val += a * this->__dat[i];
			a *= bit;
		}
	}
};

template <ull bit, ull n>
ostream &operator<<(ostream &os, const CustomBit<bit, n> &bits)
{
	os << "[";
	for (ll i = 0; i < n; ++i)
	{
		os << bits.get(i) << (i != n - 1 ? ", " : "");
	}
	os << "](bit: " << bit << ")";

	return os;
}

#pragma endregion

/* Variables */
ll N, M, Q;
ll H, W;
string dump = "";
ll codeforces_t = -1;

/* Main Function */

int main()
{
	fastio;

	ll S;
	cin >> S;

	vll ans;
	while (S > 0)
	{
		ll x = 1;
		while (x * x <= S)
			x++;

		x--;
		ans.emplace_back(x * x);
		S -= x * x;
	}

	co(ans.size());
	print(ans);

	return 0;
}

/* 文字化け注意 */