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 Rust (1.15.1) (judge verdict: Runtime Error)

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let a: usize = sc.read();
    let b: usize = sc.read();
    match solve(n, a, b) {
        Some(path) => {
            println!("YES");
            let mut ans = vec![0; 1 << n];
            let mut vis = vec![false; 1 << n];
            ans[0] = a;
            vis[a] = true;
            if dfs(0, &mut vis, &path, &mut ans, b) {
                for (i, p) in ans.into_iter().enumerate() {
                    if i > 0 {
                        print!(" ");
                    }
                    print!("{}", p);
                }
                println!();
            } else {
                unreachable!()
            }
        }
        None => println!("NO"),
    }
}

fn dfs(pos: usize, vis: &mut [bool], path: &[usize], ans: &mut [usize], b: usize) -> bool {
    if pos == ans.len() - 1 {
        return ans[pos] == b;
    }
    let cur = ans[pos];
    for i in 0..17 {
        let next = cur ^ (1 << i);
        if !vis[next] && next.count_ones() as usize == path[pos + 1] {
            ans[pos + 1] = next;
            vis[next] = true;
            if dfs(pos + 1, vis, path, ans, b) {
                return true;
            }
            vis[next] = false;
        }
    }
    false
}

fn solve(n: usize, a: usize, b: usize) -> Option<Vec<usize>> {
    let mut count: [i32; 18] = [0; 18];
    for mask in 0..(1 << n) {
        count[(mask as u32).count_ones() as usize] += 1;
    }

    let max_bit = count
        .iter()
        .enumerate()
        .filter(|&(_, &c)| c > 0)
        .map(|(i, _)| i)
        .max()
        .unwrap();

    let goal_bit = b.count_ones() as usize;

    let mut vis = [0; 18];
    let mut cur = a.count_ones() as usize;
    let mut path = vec![cur];
    vis[cur] += 1;
    if vis[cur] > count[cur] {
        return None;
    }
    while cur > 0 {
        cur -= 1;
        path.push(cur);
        vis[cur] += 1;
        if vis[cur] > count[cur] {
            return None;
        }
    }
    assert_eq!(cur, 0);
    if vis == count {
        return Some(path);
    }
    cur += 1;
    path.push(cur);
    vis[cur] += 1;
    while cur < goal_bit {
        if vis[cur - 1] > count[cur - 1] {
            return None;
        }
        if vis[cur - 1] < count[cur - 1] {
            cur -= 1;
            path.push(cur);
            vis[cur] += 1;
        } else {
            cur += 1;
            path.push(cur);
            vis[cur] += 1;
        }
        if vis[cur] > count[cur] {
            return None;
        }
    }

    while cur < max_bit {
        cur += 1;
        path.push(cur);
        vis[cur] += 1;
        if vis[cur] > count[cur] {
            return None;
        }
    }
    if vis == count {
        return Some(path);
    }
    assert_eq!(cur, max_bit);
    cur -= 1;
    path.push(cur);
    vis[cur] += 1;
    while vis != count {
        if vis[cur + 1] > count[cur + 1] {
            return None;
        }
        if vis[cur + 1] < count[cur + 1] {
            cur += 1;
            path.push(cur);
            vis[cur] += 1;
        } else {
            cur -= 1;
            path.push(cur);
            vis[cur] += 1;
        }
        if vis[cur] > count[cur] {
            return None;
        }
    }
    if path[path.len() - 1] != goal_bit {
        None
    } else {
        Some(path)
    }
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}