/**
 * Author: Sergey Kopeliovich (Burunduk30@gmail.com)
 */

// Egor Suvorov: sometimes u need g++ -static to fix some random troubles
#define VERSION "0.1.9"

#include <cassert>
#include <cstdio>
#include <algorithm>

/** Fast allocation */

#ifdef FAST_ALLOCATOR_MEMORY
	int allocator_pos = 0;
	char allocator_memory[(int)FAST_ALLOCATOR_MEMORY];
	void * operator new (size_t n) {
		// fprintf(stderr, "n=%ld\n", n);
		char *res = allocator_memory + allocator_pos;
		assert(n <= (size_t)((int)FAST_ALLOCATOR_MEMORY - allocator_pos));
		allocator_pos += n;
		return (void *)res;
	}
	void operator delete (void *) noexcept { }
	void operator delete (void *, size_t) noexcept { }
	//inline void * operator new [] (size_t) { assert(0); }
	//inline void operator delete [] (void *) { assert(0); }
#endif

/** Fast input-output */

template <class T = int> inline T readInt();						
inline double readDouble();
inline int readUInt();					 
inline int readChar(); // first non-blank character
inline void readWord(char *s); 
inline bool readLine(char *s); // do not save '\n'
inline bool isEof();
inline int getChar(); 
inline int peekChar();
inline bool seekEof();
inline void skipBlanks();

template <class T> inline void writeInt(T x, char end = 0, int len = -1);
inline void writeChar(int x); 
inline void writeWord(const char *s);
inline void writeDouble(double x, int len = 0); // works correct only for |x| < 2^{63}
inline void flush();

static struct buffer_flusher_t {
	~buffer_flusher_t() {
		flush();
	}
} buffer_flusher;

/** Read */

static const int buf_size = 4096;

static unsigned char buf[buf_size];
static int buf_len = 0, buf_pos = 0;

inline bool isEof() {
	if (buf_pos == buf_len) {
		buf_pos = 0, buf_len = fread(buf, 1, buf_size, stdin);
		if (buf_pos == buf_len)
			return 1;
	}
	return 0;
}

inline int getChar() { 
	return isEof() ? -1 : buf[buf_pos++];
}

inline int peekChar() { 
	return isEof() ? -1 : buf[buf_pos];
}

inline bool seekEof() { 
	int c;
	while ((c = peekChar()) != -1 && c <= 32)
		buf_pos++;
	return c == -1;
}

inline void skipBlanks() {
	while (!isEof() && buf[buf_pos] <= 32U)
		buf_pos++;
}

inline int readChar() {
	int c = getChar();
	while (c != -1 && c <= 32)
		c = getChar();
	return c;
}

inline int readUInt() {
	int c = readChar(), x = 0;
	while ('0' <= c && c <= '9')
		x = x * 10 + (c - '0'), c = getChar();
	return x;
}

template<class T>
inline T readInt() {
	int minus = 0, c = readChar();
	T x = 0;
	if (c == '-')
		minus = 1, c = getChar();
	else if (c == '+')
		c = getChar();
	for (; '0' <= c && c <= '9'; c = getChar())
		if (minus)
			x = x * 10 - (c - '0'); // take care about -2^{31}
		else
			x = x * 10 + (c - '0');
	return x;
}

inline double readDouble() {
	int s = 1, c = readChar();
	double x = 0;
	if (c == '-')
		s = -1, c = getChar();
	while ('0' <= c && c <= '9')
		x = x * 10 + (c - '0'), c = getChar();
	if (c == '.') {
		c = getChar();
		double coef = 1;
		while ('0' <= c && c <= '9')
			x += (c - '0') * (coef *= 1e-1), c = getChar();
	}
	return s == 1 ? x : -x;
}

inline void readWord(char *s) { 
	int c = readChar();
	while (c > 32)
		*s++ = c, c = getChar();
	*s = 0;
}

inline bool readLine(char *s) { 
	int c = getChar();
	while (c != '\n' && c != -1)
		*s++ = c, c = getChar();
	*s = 0;
	return c != -1;
}

/** Write */

static int write_buf_pos = 0;
static char write_buf[buf_size];

inline void writeChar(int x) {
	if (write_buf_pos == buf_size)
		fwrite(write_buf, 1, buf_size, stdout), write_buf_pos = 0;
	write_buf[write_buf_pos++] = x;
}

inline void flush() {
	if (write_buf_pos) {
		fwrite(write_buf, 1, write_buf_pos, stdout), write_buf_pos = 0;
		fflush(stdout);
	}
}

template<class T> 
inline void writeInt(T x, char end, int output_len) {
	if (x < 0)
		writeChar('-');

	char s[24];
	int n = 0;
	while (x || !n) 
		s[n++] = '0' + abs((int)(x % 10)), x /= 10;  // `abs`: take care about -2^{31}
	while (n < output_len)
		s[n++] = '0';
	while (n--)
		writeChar(s[n]);
	if (end)
		writeChar(end);
}

inline void writeWord(const char *s) {
	while (*s)
		writeChar(*s++);
}

inline void writeDouble(double x, int output_len) {
	if (x < 0)
		writeChar('-'), x = -x;
	assert(x <= (1LLU << 63) - 1);
	long long t = (long long)x;
	writeInt(t), x -= t;
	writeChar('.');
	for (int i = output_len - 1; i > 0; i--) {
		x *= 10;
		t = std::min(9, (int)x);
		writeChar('0' + t), x -= t;
	}
	x *= 10;
	t = std::min(9, (int)(x + 0.5));
	writeChar('0' + t);
}