
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
For which positive integers \(n\) do there exist a positive integer \(m\) and positive integers \(a_1, a_2, \cdots, a_{m-1}\) such that
\[ n = a_1(m - a_1) + a_2(m - a_2) + \cdots + a_{m-1}(m - a_{m-1}), \]
where \(a_1, a_2, \cdots, a_{m-1}\) are distinct, and \(1 \leq a_i \leq m - 1\) for \(i=1,2,\cdots,m-1\)?

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