Problem Statement

Note: The output of this problem contains non-ASCII characters.
The non-ASCII characters contained are as follows: ∧∨¬⊥⊤

For $N$ atomic formulas $P_1, P_2, \cdots, P_N$, consider the following truth table.

$ \large \begin{array}{cccc|c} P_1 & P_2 & \cdots & P_N & A \\ \hline Q_{1,1} & Q_{1,2} & \cdots & Q_{1,N} & R_1 \\ Q_{2,1} & Q_{2,2} & \cdots & Q_{2,N} & R_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ Q_{2^N,1} & Q_{2^N,2} & \cdots & Q_{2^N, N} & R_{2^N} \end{array} $

$Q_{i,j}$ is $1$ or $0$, where 1 represents true and 0 represents false.
Also, $R_i$ is $1$ or $0$, and represents true or false in the same way.
Output a logical formula $A$ that is valid for the truth function represented by this truth table.

Input

$N$
$Q_{1,1} \, Q_{1,2} \, \cdots \, Q_{1,N} \, R_1$
$Q_{2,1} \, Q_{2,2} \, \cdots \, Q_{2,N} \, R_2$
$\vdots$
$Q_{2^n,1} \, Q_{2^n,2} \, \cdots \, Q_{2^n,N} \, R_{2^N}$

$1 \le N \le 12$
Each of $Q_{i,j}, R_i(1 \le i \le 2^N,\, 1 \le j \le N)$ is $0$ or $1$.

Output

Output a logical formula $A$ that is valid for the given truth function. The output must follow the format below.

• Start with "A=".
• If $P_i$ is true, output $P_i$; if it is false, output $\lnot P_i$.
• For every i, $P_i$ is represented in the output as "P_i". (Regardless of how many digits i has; for example, when i=12 it is represented as P_12.)
• If $R_i$ is true, include in the logical formula A a formula of the form "($Q_1$∧$Q_2$∧$\cdots$∧$Q_N$)" based on the i-th row of the truth table.
• Here, $Q_i$ denotes "P_i" if $P_i$ is true, and otherwise $Q_i$ denotes "¬P_i".
• Letting $X_1, X_2, \cdots, X_N$ be the formulas included here, the logical formula $A$ must be represented in the form "A=$X_1$∨$X_2$∨$\cdots$∨$X_N$".
• When $i > j$, $X_j$ must appear before $X_i$ in the logical formula A.
• When $i > j$, $Q_j$ must appear before $Q_i$ in the logical formula $X_k$.
• If $R_i$ is true for all $i$, the logical formula A is represented as "A=⊤". (Not T. ⊤ is read as top.)
• If $R_i$ is false for all $i$, the logical formula A is represented as "A=⊥". (⊥ represents a contradiction. It is read as bottom.)

Sample

2
1 1 0
1 0 1
0 1 1
0 0 0

A=(P_1∧¬P_2)∨(¬P_1∧P_2)

1
0 0
1 0

A=⊥

2
1 1 1
1 0 1
0 1 1
0 0 1

A=⊤

4
1 1 1 1 0
1 1 1 0 1
1 1 0 1 0
1 1 0 0 1
1 0 1 1 0
1 0 1 0 1
1 0 0 1 0
1 0 0 0 1
0 1 1 1 0
0 1 1 0 1
0 1 0 1 0
0 1 0 0 1
0 0 1 1 0
0 0 1 0 1
0 0 0 1 0
0 0 0 0 1

A=(P_1∧P_2∧P_3∧¬P_4)∨(P_1∧P_2∧¬P_3∧¬P_4)∨(P_1∧¬P_2∧P_3∧¬P_4)∨(P_1∧¬P_2∧¬P_3∧¬P_4)∨(¬P_1∧P_2∧P_3∧¬P_4)∨(¬P_1∧P_2∧¬P_3∧¬P_4)∨(¬P_1∧¬P_2∧P_3∧¬P_4)∨(¬P_1∧¬P_2∧¬P_3∧¬P_4)

2
1 1 1
1 0 1
0 1 0
0 0 0

A=(P_1∧P_2)∨(P_1∧¬P_2)