Problem:
Score : 800 points

Problem Statement

You are given integers N,\ A and B.
Determine if there exists a permutation (P_0,\ P_1,\ ...\ P_{2^N-1}) of (0,\ 1,\ ...\ 2^N-1) that satisfies all of the following conditions, and create one such permutation if it exists.

P_0=A

P_{2^N-1}=B

For all 0 \leq i < 2^N-1, the binary representations of P_i and P_{i+1} differ by exactly one bit.

Constraints

1 \leq N \leq 17

0 \leq A \leq 2^N-1

0 \leq B \leq 2^N-1

A \neq B

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N A B

Output

If there is no permutation that satisfies the conditions, print NO.

If there is such a permutation, print YES in the first line.
Then, print (P_0,\ P_1,\ ...\ P_{2^N-1}) in the second line, with spaces in between.
If there are multiple solutions, any of them is accepted.

Sample Input 1

2 1 3

Sample Output 1

YES
1 0 2 3

The binary representation of P=(1,0,2,3) is (01,00,10,11), where any two adjacent elements differ by exactly one bit.

Sample Input 2

3 2 1

Sample Output 2

NO

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>
#include <array>
#include <list>
#include <set>
#include <map>
#include <utility>
#include <string>
#include <sstream>
#include <algorithm>
#include <random>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <ctime>
#include <cmath>
#include <cassert>
#include <climits>

using namespace std;

int main()
{
	uint32_t n, a, b; cin >> n >> a >> b;

	uint32_t d = a ^ b;
	int bits = 0;

	{
		uint32_t cur_d = d;
		for (int i = 0; i < 32; i++) {
			if (cur_d & 1U) bits++;
			cur_d >>= 1;
		}
	}
	
	bool is_ok = true;
	if ((bits & 1) == 0) {
		is_ok = false;
	}

	if (!is_ok) {
		cout << "NO" << endl;
		return 0;
	}

	cout << "YES" << endl;

	cout << a;

	uint32_t prev = a;

	for (int i = 0; i < ((1U << n) - 2); i++) {
		if (d != 0) {
			for (int j = 0; j < 32; j++) {
				if (d & (1U << j)) {
					prev = prev ^ (d & (1U << j));
					d    = d ^ (1U << j);
					break;
				}
			}
		}
		else {
			prev = prev ^ 1;
		}
		cout << " " << prev;
	}

	cout << " " << b << endl;
	cout.flush();

	return 0;
}