
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]


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On a unit circle centered at point \( O \), five points \( A_{1}, A_{2}, \cdots, A_{5} \) are taken sequentially; let \( P \) be a point inside the circle. Denote the intersection point of the line segment \( A_{i} A_{i+2} \) with the line segment \( P A_{i+1} \) as \( Q_{i} \) for \( i = 1, 2, \cdots, 5 \), where \( A_{6} = A_{1} \). Let \( O Q_{i} = d_{i} \) for \( i = 1, 2, \cdots, 5 \). Find the product \( A_{1} Q_{1} \cdot A_{2} Q_{2} \cdot \cdots \cdot A_{5} Q_{5} \).

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