
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  $ m > 1$  be an integer,  $ n$  is an odd number satisfying  $ 3\le n < 2m,$  number  $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$  satisfies  $ (1)$  for any  $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$  is a permutation of  $ 1,2,3,\cdots,m; (2)$  for any  $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1$  holds. Find the minimal value of  $ M$ , where  $ M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.$ 

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