
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 \( \alpha_{i} \in\left(0, \frac{\pi}{2}\right) \) for \( i=1, 2, 3, \ldots, n \) where \( n \geq 2 \), and that \( \prod_{i=1}^{n} \sin \alpha_{i} = a \) (a constant), find the maximum value of \( \prod_{i=1}^{n} \cos \alpha_{i} \).

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