
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
In a right triangle \( \triangle ABC \), let \( BC = a \). If a segment \( PQ \) of length \( 2a \) has a midpoint at point \( A \), what should be the angle between the vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{BC} \) to maximize the value of \( \overrightarrow{BP} \cdot \overrightarrow{CQ} \)? Also, find this maximum value.

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