Migrated from Lutece 821 G(x)

All parts of this problem, including description, images, samples, data and checker, might be broken. If you find bugs in this problem, please contact the admins.

Description

For a binary number $x$ with $n$ digits ($A_{n}A_{n-1}A_{n-2}\cdots A_{2}A_{1}$), we encode it as $G(x) = x\otimes \lfloor\frac{x}{2}\rfloor$. Where $\otimes$ is bitwise XOR operation and $\lfloor x\rfloor$ indicates the largest integer which is not greater than $x$.

Due to some reasons, Mzry1992 encode his password $P$ into $G(P)$, and additionally, he encode $P + 1$ into $G(P + 1)$ too, and write $G(P)$ and $G(P + 1)$ into his diary.

This story happened many years ago and now you hold the diary with these numbers in your hands. Unfortunately, some digits are unreadable now. Could you determine the values of these digits using the readable digits?

Input

The first line has a number $T$ ($T\leq 100$) , indicating the number of test cases.

For every test case, it has $2$ lines of same number of digits describe $G(P)$ and $G(P + 1)$, In every line, it only contains 1, 0 and ?. Unreadable digits are denoted with symbol ?, The length of every line in the input is up to $10^5$.

Output

For every case, you should output Case #t: at first, without quotes. The $t$ is the case number starting from $1$.

Then, if there is impossible to restore $G(P)$ and $G(P + 1)$, you should output Impossible in the second line.

Otherwise, if $G(P)$ is unique, you should output restored $G(P)$ and $G(P + 1)$ in the same format.

Otherwise, you should output Ambiguous and the number of possible $G(P)$ in the second line.

Samples

3
10??
10??
0010
0110
1?01
0?01

Case #1:
Ambiguous 3
Case #2:
0010
0110
Case #3:
Impossible

Note

In the first sample case, the three possible situations are:

• $1010$ $1011$
• $1011$ $1001$
• $1001$ $1000$

Resources

2013 ACM/ICPC Asia Regional Chengdu Online