Problem Statement

Output all length-$N$ sequences $A=(A_1,A_2,\ldots,A_N)$ consisting of non-negative integers that satisfy $\displaystyle\sum_{i=1}^{N} i\times A_i=K$, in lexicographic order from smallest to largest.

Here, you will only receive inputs such that the number of sequences satisfying the condition is at most $3\times10 ^ 5$.

What is lexicographic order for sequences?

A sequence $S = (S_1,S_2,\ldots,S_{|S|})$ is said to be lexicographically smaller than a sequence $T = (T_1,T_2,\ldots,T_{|T|})$ if either 1. or 2. below holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively.

1. $|S| \lt |T|$ and $(S_1,S_2,\ldots,S_{|S|}) = (T_1,T_2,\ldots,T_{|S|})$.
2. There exists an integer $1 \leq i \leq \min\lbrace |S|, |T| \rbrace$ such that both of the following hold.
   • $(S_1,S_2,\ldots,S_{i-1}) = (T_1,T_2,\ldots,T_{i-1})$
   • $S_i$ is (numerically) smaller than $T_i$.

Constraints

• $1\le N\le 10$
• $1\le K\le 2\times10 ^ 5$
• The number of sequences satisfying the condition is at most $3\times10 ^ 5$.
• All input values are integers.

Input

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

• $N$ $K$

Output

Let $q$ be the number of sequences of non-negative integers satisfying the condition; output them over $q$ lines. Each line should contain the elements of a sequence of non-negative integers satisfying the condition, in order, separated by spaces. For every sequence, all sequences outputed before it must be lexicographically smaller than it.

3 8

0 1 2
0 4 0
1 2 1
2 0 2
2 3 0
3 1 1
4 2 0
5 0 1
6 1 0
8 0 0

For example, for the sequence $(0,1,2)$ we have $0\times1+1\times2+2\times3=0+2+6=8$, so it satisfies the condition. There is no sequence satisfying the condition that is lexicographically smaller than this one, so output 0 1 2 on the first line.

Including $(0,1,2)$, ten sequences satisfy the condition. Output these in lexicographic order from smallest to largest.

1 200000

200000

8 9

0 0 0 1 1 0 0 0
0 0 1 0 0 1 0 0
0 0 3 0 0 0 0 0
0 1 0 0 0 0 1 0
0 1 1 1 0 0 0 0
0 2 0 0 1 0 0 0
0 3 1 0 0 0 0 0
1 0 0 0 0 0 0 1
1 0 0 2 0 0 0 0
1 0 1 0 1 0 0 0
1 1 0 0 0 1 0 0
1 1 2 0 0 0 0 0
1 2 0 1 0 0 0 0
1 4 0 0 0 0 0 0
2 0 0 0 0 0 1 0
2 0 1 1 0 0 0 0
2 1 0 0 1 0 0 0
2 2 1 0 0 0 0 0
3 0 0 0 0 1 0 0
3 0 2 0 0 0 0 0
3 1 0 1 0 0 0 0
3 3 0 0 0 0 0 0
4 0 0 0 1 0 0 0
4 1 1 0 0 0 0 0
5 0 0 1 0 0 0 0
5 2 0 0 0 0 0 0
6 0 1 0 0 0 0 0
7 1 0 0 0 0 0 0
9 0 0 0 0 0 0 0

• Source: AtCoder ABC 473 D