Broken Device

Anna and Bruno are archaeologists. They are investigating ruins in Iran.

Their tasks are as follows: Anna visits the ruins and discovers artifacts, and Bruno analyzes the results at the base camp.

Their investigation is scheduled for $Q$ (= 1000) days. Every day, Anna sends the day's result to Bruno with a communication device. The result for each day is an integer $X$.

Anna can use the communication device once per day only. It can send a sequence of length $N$ (= 150) consisting of 0s and 1s.

However, the device is broken. The sequence of length $N$ has several broken positions. At a broken position, the device always sends 0, regardless of the value Anna sets. Anna can see the broken positions when she sends a sequence, but Bruno does not know them. The broken positions and their number can change every day.

There is a danger that the investigation will be delayed. You are a candidate for the international programming contest in Iran, so Anna and Bruno ask you to write a program that sends the results of their investigation.

Write two programs that make the communication between Anna and Bruno work.

• The first program takes the sequence length $N$, the integer to send $X$, the number of broken positions $K$, and the broken positions $P$, and sets the sequence $S$ that Anna sends.
• The second program takes the sequence $A$ that Bruno receives and recovers the integer $X$.

At a working position, $S$ and $A$ have the same value. At a broken position, $A$ always has the value 0 regardless of the value of $S$.

The sample grader reads the following data from standard input.

• The first line contains an integer $Q$.
• Then, the information of $Q$ queries is given.
• The information of each query consists of two lines.
  • The first line contains three space-separated integers $N$, $X$, $K$. This means the length of the sequence to send is $N$, the integer Anna sends is $X$, and there are $K$ broken positions.
  • The second line contains $K$ space-separated integers $P_0, P_1, \cdots, P_{K-1}$. This means that for each $i$ ($0 \le i \le K - 1$), the $P_i$-th position in the sequence is broken.

When the program terminates successfully, the sample grader writes the following information to standard output. (Quotation marks are not actually written.)

• If your program is judged as Wrong Answer, the grader writes the type in the form Wrong Answer [1], and your program is terminated.
• If no call to Anna is judged as Wrong Answer, the grader writes Accepted and the value of $L^*$. For the value of $L^*$, see Scoring.

If your program falls under several types of Wrong Answer, the grader reports only one of them.

• $Q = 1000$.
• $N = 150$.
• $0 \le X \le 10^{18}$.
• $1 \le K \le 40$.
• $0 \le P_i \le N - 1$ ($0 \le i \le K - 1$).
• $P_i < P_{i+1}$ ($0 \le i \le K - 2$).