Problem Statement

Consider Alice communicating a sequence $A_{1}, \dots, A_{N}$ of length $N$ to Bob. Here, $1 \leq A_{1} < \dots < A_{N} \leq 10^{9}$ holds.

Alice chooses an integer $K$ between $1$ and $N$, inclusive, and can communicate to Bob only $K$ non-empty 01 strings $S_{1}, \dots, S_{K}$. However, using the given integer $Q (= 377)$, the total string length $\sum_{i=1}^{K} |S_i|$ must be at most $Q$.

Bob wants to restore the sequence $A$ given to Alice, using only $N, Q$ and the strings $S_{1}, \dots, S_{K}$ communicated by Alice.

Write programs for Alice and Bob.

Note that partial scores are awarded for the cases $Q=420, 410$.

Input and Output

Your program should first receive input in the following format.

$ \mathrm{Player} $
$ N\ Q $

Here, $\mathrm{Player}$ is the string Alice or the string Bob, $N$ is the length of the sequence $A$, and $Q$ is the upper limit on the total length of the strings Alice can output. ( $1 \leq N \leq 14$, $Q \in \{377, 410, 420\}$)

• When $\mathrm{Player} = $ Alice, your program must act as Alice from then on.
• When $\mathrm{Player} = $ Bob, your program must act as Bob from then on.

For Alice

Receive the sequence $A$ ( $1 \leq A_{1} < \dots < A_{N} \leq 10^{9}$ ) in the following format.

$A_{1}\ A_{2}\ \dots\ A_{N}$

Then, output the $K$ strings $S_{1}, \dots, S_{K}$ to communicate to Bob, one per line, in the following format. Here, $1 \leq K \leq N$, each string must consist only of 0 and 1, and the total string length $\sum_{i=1}^{K} |S_{i}|$ must be at most $Q$.

$K$
$S_1$
$S_2$
$\vdots$
$S_K$

For Bob

Receive Alice's output $S_{1}, \dots, S_{K}$ in the following format.

$K$
$S_1$
$S_2$
$\vdots$
$S_K$

Then, restore the sequence $A$ of length $N$ that was given to Alice as input, and output it separated by spaces in the following format.

$A_{1}\ A_{2}\ \dots\ A_{N}$

Notes

• Each time you output, append a newline at the end and flush standard output.
• The judge's result for invalid output is undefined. Note that you may receive TLE or RE rather than WA.

Sample

Sample 1

When $A=(2, 3, 5)$

Alice's input

Alice
3 380
2 3 5

Alice's output

3
00
111
00000

Bob's input

Bob
3 380
3
00
111
00000

Bob's output

2 3 5