
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 two equiangular polygons \(P\) and \(Q\) with different numbers of sides; each angle of \(P\) is \(p\) degrees, and each angle of \(Q\) is \(q\) degrees, determine the number of possibilities for the pair \((p, q)\) where either \(p = 2q\) or \(q = 2p\).

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