Score : $400$ points

Problem Statement

Let $N$ be a positive integer. Define the imbalance of a sequence $A=(A\_1, A\_2, \dots, A\_{2^N})$ of non-negative integers of length $2^N$ as the non-negative integer value obtained by the following operation:

• Initially, set $X=0$.
• Perform the following series of operations $N$ times:
  • Update $X$ to $\max(X, \max(A) - \min(A))$, where $\max(A)$ and $\min(A)$ denote the maximum and minimum values of sequence $A$, respectively.
  • Form a new sequence of half the length by pairing elements from the beginning two by two and arranging their sums. That is, set $A \gets (A\_1 + A\_2, A\_3 + A\_4, \dots, A\_{\vert A \vert - 1} + A\_{\vert A \vert})$.
• The final value of $X$ is the imbalance.

For example, when $N=2, A=(6, 8, 3, 5)$, the imbalance is $6$ through the following steps:

• Initially, $X=0$.
• The first series of operations is as follows:
  • Update $X$ to $\max(X, \max(A) - \min(A)) = \max(0, 8 - 3) = 5$.
  • Set $A$ to $(6+8, 3+5) = (14, 8)$.
• The second series of operations is as follows:
  • Update $X$ to $\max(X, \max(A) - \min(A)) = \max(5, 14 - 8) = 6$.
  • Set $A$ to $(14 + 8) = (22)$.
• Finally, $X=6$.

You are given a non-negative integer $K$. Among all sequences of non-negative integers of length $2^N$ with sum $K$, construct a sequence that minimizes the imbalance.

Constraints

• $1 \leq N \leq 20$
• $0 \leq K \leq 10^9$
• $N$ and $K$ are integers.

Input

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

$N$ $K$

Output

Let $B=(B\_1,B\_2,\dots,B\_{2^N})$ be a sequence with minimum imbalance. Let $U$ be the imbalance of $B$. Output a solution in the following format:

$U$
$B_1$ $B_2$ $\dots$ $B_{2^N}$

If there are multiple solutions, any of them will be considered correct.

$(5, 6)$ is a sequence with imbalance $1$, which is the minimum imbalance among sequences satisfying the condition.

Samples

1 11

1
5 6

3 56

0
7 7 7 7 7 7 7 7