
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $n$  be a fixed positive integer. Let  $$ A=\begin{bmatrix} a_{11} & a_{12} & \cdots &a_{1n}   a_{21} & a_{22} & \cdots &a_{2n}   \vdots & \vdots & \cdots & \vdots  a_{n1} & a_{n2} & \cdots &a_{nn} \end{bmatrix}\quad
\text{and} \quad B=\begin{bmatrix} b_{11} & b_{12} & \cdots &b_{1n}   b_{21} & b_{22} & \cdots &b_{2n}   \vdots & \vdots & \cdots & \vdots  b_{n1} & b_{n2} & \cdots &b_{nn} \end{bmatrix}\quad $$  be two  $n\times n$  tables such that  $\{a_{ij}|1\le i,j\le n\}=\{b_{ij}|1\le i,j\le n\}=\{k\in N^*|1\le k\le n^2\}$ .
One can perform the following operation on table  $A$ : Choose  $2$  numbers in the same row or in the same column of  $A$ , interchange these  $2$  numbers, and leave the remaining  $n^2-2$  numbers unchanged. This operation is called a **transposition** of  $A$ .
Find, with proof, the smallest positive integer  $m$  such that for any tables  $A$  and  $B$ , one can perform at most  $m$  transpositions such that the resulting table of  $A$  is  $B$ .

Present the answer in LaTex format: \boxed{Your answer}
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