
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
The right triangular prism \(ABC-A_1B_1C_1\) has a base \(\triangle ABC\) which is an equilateral triangle. Points \(P\) and \(E\) are movable points (including endpoints) on \(BB_1\) and \(CC_1\) respectively. \(D\) is the midpoint of side \(BC\), and \(PD \perp PE\). Find the angle between lines \(AP\) and \(PE\).

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