
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 concentric circles with radii \( r \) and \( R \) (\( r < R \)). A line is drawn through a point \( P \) on the smaller circle, intersecting the larger circle at points \( B \) and \( C \). The perpendicular to \( BC \) at point \( P \) intersects the smaller circle at point \( A \). Find \( |PA|^2 + |PB|^2 + |PC|^2 \).

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