
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
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Evaluate the summation \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\).

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