
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 \( n \geq 3 \), let \( d_{1}, d_{2}, \ldots, d_{n} \) be positive integers whose greatest common divisor is 1, and each \( d_{i} \) divides \( d_{1} + d_{2} + \cdots + d_{n} \) for \( i = 1, 2, \ldots, n \). Find the smallest positive integer \( k \) such that \( d_{1} d_{2} \cdots d_{n} \) divides \((d_{1} + d_{2} + \cdots + d_{n})^{k}\).

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