201271: [AtCoder]ARC127 B - Ternary Strings

Problem Statement

Given are integers $N$ and $L$. Find a tuple of $3N$ strings $(S_1,S_2,\cdots,S_{3N})$ that satisfies all of the following conditions.

• $S_i$ is a string of length $L$ consisting of 0, 1, 2.

• All $S_i$ are pairwise distinct.

• For every $j$ ($1 \leq j \leq L$) and every $c=$0, 1, 2, the following holds.

  • For exactly $N$ of the strings $S_i$, the $j$-th character is $c$.

• Let $t$ be the lexicographically largest string among $S_1,S_2,\cdots,S_{3N}$. $t$ for this tuple is the lexicographically smallest among all strings that $t$ can be.

Constraints

• $1 \leq N \leq 5 \times 10^4$
• $1 \leq L \leq 15$
• $3N \leq 3^L$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $L$

Output

Print the answer in the following format:

$S_1$
$S_2$
$\vdots$
$S_{3N}$

If there are multiple solutions satisfying the conditions, any of them will be accepted.

Sample Input 1

2 2

Sample Output 1

00
02
11
12
20
21

This Sample Output satisfies all conditions.

For example, there are two strings whose second character is 0.

Also, we have $t=$21 in this sample, and $t$ is never lexicographically smaller than this.