
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 in a plane with radii \( R \) and \( r \) (\( R > r \)). Point \( P \) is a fixed point on the smaller circle, and point \( B \) moves along the larger circle. Line \( BP \) intersects the larger circle again at point \( C \). The perpendicular line to \( BP \) at \( P \), denoted by \( l \), intersects the smaller circle again at \( A \) (if \( l \) is tangent to the circle at \( P \), then \( A = P \)).

I. Determine the set of values taken by \( B C^2 + C A^2 + A B^2 \).

II. Determine the geometric locus of the midpoint of segment \( AB \).

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