Problem: We have a pyramid with N steps, built with blocks. The steps are numbered 1 through N from top to bottom. For each 1≤i≤N, step i consists of 2i-1 blocks aligned horizontally. The pyramid is built so that the blocks at the centers of the steps are aligned vertically.

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A pyramid with N=4 steps

Snuke wrote a permutation of (1, 2, ..., 2N-1) into the blocks of step N. Then, he wrote integers into all remaining blocks, under the following rule:

* The integer written into a block b must be equal to the median of the three integers written into the three blocks directly under b, or to the lower left or lower right of b.



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Writing integers into the blocks

Afterwards, he erased all integers written into the blocks. Now, he only remembers that the integer written into the block of step 1 was x.

Construct a permutation of (1, 2, ..., 2N-1) that could have been written into the blocks of step N, or declare that Snuke's memory is incorrect and such a permutation does not exist.

Constraints

* 2≤N≤10^5
* 1≤x≤2N-1

Input

The input is given from Standard Input in the following format:


N x


Output

If no permutation of (1, 2, ..., 2N-1) could have been written into the blocks of step N, print `No`.

Otherwise, print `Yes` in the first line, then print 2N-1 lines in addition.

The i-th of these 2N-1 lines should contain the i-th element of a possible permutation.

Examples

Input

4 4


Output

Yes
1
6
3
7
4
5
2


Input

2 1


Output

No

Solution (in JAVA):

Solution:

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.List;
import java.util.StringTokenizer;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

public class Main {
    public static void main(final String[] args) {
        final FastScanner scanner = new FastScanner(System.in);
        final int n = scanner.nextInt();
        final int x = scanner.nextInt();

        if (x == n) {
            System.out.println("Yes");
            final String answer = IntStream.rangeClosed(1, 2 * n - 1)
                    .mapToObj(Integer::toString)
                    .collect(Collectors.joining(" "));
            System.out.println(answer);
        } else if (x >= 3 && x <= 2 * n - 3) {
            System.out.println("Yes");
            final List<Integer> list = IntStream.rangeClosed(1, 2 * n - 1)
                    .filter(i -> !(i >= x - 2 && i <= x + 2))
                    .boxed()
                    .collect(Collectors.toList());
            list.add(n - 3, x + 2);
            list.add(n - 2, x - 1);
            list.add(n - 1, x);
            list.add(n, x + 1);
            list.add(n + 1, x - 2);
            final String answer = list.stream()
                    .map(i -> Integer.toString(i))
                    .collect(Collectors.joining(" "));
            System.out.println(answer);
        } else if (x == 2) {
            System.out.println("Yes");
            final List<Integer> list = IntStream.rangeClosed(5, 2 * n - 1)
                    .boxed()
                    .collect(Collectors.toList());
            list.add(n - 3, 4);
            list.add(n - 2, 1);
            list.add(n - 1, 2);
            list.add(n, 3);
            final String answer = list.stream()
                    .map(i -> Integer.toString(i))
                    .collect(Collectors.joining(" "));
            System.out.println(answer);
        } else if (x == 2 * n - 2) {
            System.out.println("Yes");
            final List<Integer> list = IntStream.rangeClosed(1, 2 * n - 5)
                    .boxed()
                    .collect(Collectors.toList());
            list.add(n - 3, 2 * n - 4);
            list.add(n - 2, 2 * n - 1);
            list.add(n - 1, 2 * n - 2);
            list.add(n, 2 * n - 3);
            final String answer = list.stream()
                    .map(i -> Integer.toString(i))
                    .collect(Collectors.joining(" "));
            System.out.println(answer);
        } else {
            System.out.println("No");
        }
    }

    private static class FastScanner {
        private final BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(final InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (final IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }
    }
}