
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]


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For given integer  $n \geq 3$ , set  $S =\{p_1, p_2, \cdots, p_m\}$  consists of permutations  $p_i$  of  $(1, 2, \cdots, n)$ . Suppose that among every three distinct numbers in  $\{1, 2, \cdots, n\}$ , one of these number does not lie in between the other two numbers in every permutations  $p_i$  ( $1 \leq i \leq m$ ). (For example, in the permutation  $(1, 3, 2, 4)$ ,  $3$  lies in between  $1$  and  $4$ , and  $4$  does not lie in between  $1$  and  $2$ .) Determine the maximum value of  $m$ .

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