
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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A fair die with eight sides labeled 1, 2, 3, 5, 7, 10, 14, 15 is rolled four times. Calculate the probability that the product of the four numbers rolled is a perfect square. Express your answer as a fraction $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, and find $p+q$.

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