
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
Given a positive integer \( n \geq 3 \), find the smallest positive integer \( m \) such that any arbitrary partition of the set \( S = \{1, 2, \ldots, m\} \) into two non-empty disjoint subsets contains a subset with \( n \) numbers \( x_1, x_2, \ldots, x_n \) (with repetition allowed) where \( x_1 + x_2 + \cdots + x_{n-1} = x_n \).

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