
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(\ge2) $  be a positive integer. Find the minimum  $ m $ , so that there exists  $x_{ij}(1\le i ,j\le n)$  satisfying:
(1)For every  $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $  or  $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$ (2)For every  $1\le i \le n$ , there are at most  $m$  indices  $k$  with  $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$ (3)For every  $1\le j \le n$ , there are at most  $m$  indices  $k$  with  $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$ 

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