
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 $U=\{1,2,3, \cdots, n\}$ where $n \geq 3$, and let $S$ be a subset of $U$. If an element that does not belong to $S$ appears somewhere in a permutation $P$ of $U$, and in this permutation $P$ it is positioned between two elements that belong to $S$, then we say that $S$ is split by the permutation $P$ (for example, the permutation 13542 can split the subset $\{1,2,3\}$, but cannot split the subset $\{3,4,5\}$). Determine the maximum positive integer $m$ such that for any $m$ subsets of $U$ (each subset having at least 2 elements and at most $n-1$ elements), there exists a permutation of $U$ that splits all these $m$ subsets simultaneously.

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