An antique machine with $\binom{N}{3}$ switches capable of processing integers in the range $0 \ldots 2^N - 1$ has just been discovered. Each switch is associated to a distinct integer in $0 \ldots 2^N - 1$ with exactly three ones in its binary representation. By setting switches associated with number $X_0, X_1, \ldots, X_{M-1}$ to on, any integer $Y$ passing through the machine will render a result of $Y \oplus X_0 \oplus X_1 \oplus \ldots \oplus X_{M-1}$ (here “$\oplus$” stands for bitwise- XOR).

Further inspections reveal that contrary to what we assumed in problem B, some of the switches on the machine are damaged due to their old age. We are interested in whether a configuration transforming integer $S$ into $T$ still exists, and if so, the minimum number of switches that have to be set to on to make it possible.

WARNING: a naïve algorithm might not be sufficient to solve this problem.

Input

There are multiple test cases in the input file.

Each test case starts with two integers, $N$ and $M$ ($1 \le N \le 20$), representing the number of bits and the number of functioning switches, respectively. Two integers, $S$ and $T$ ($0 \le S, T < 2^N$), come in the next line, followed by another $M$ lines, the i^th one describing the value $V_i$ associated to the i^th switch ($0 \le V_i < 2^N$).

Two successive test cases are separated by a blank line. A case with $N = 0$ and $M = 0$ indicates the end of the input file, and should not be processed by your program.

Output

For each test case, please print a single integer, the minimum number of operations, or “Impossible” (without quotes) if no feasible sequence exists.

Example

Sample Input
6 7
55 21
11
22
25
56
41
49
28

5 2
0 21
22
28

0 0

Output for the Sample Input
Case #1: 2
Case #2: Impossible