
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 an integer \( n \geq 2 \), for any pairwise coprime positive integers \( a_1, a_2, \ldots, a_n \), let \( A = a_1 + a_2 + \ldots + a_n \). Denote by \( d_i \) the greatest common divisor (gcd) of \( A \) and \( a_i \) for \( i = 1, 2, \ldots, n \). Denote by \( D_i \) the gcd of the remaining \( n-1 \) numbers after removing \( a_i \). Find the minimum value of \( \prod_{i=1}^{n} \frac{A - a_i}{d_i D_i} \).

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