
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
Consider two concentric circles on the same plane, with radii \( R \) and \( r \) respectively (where \( R > r \)). Let \( P \) be a fixed point on the circumference of the smaller circle, and \( B \) be a moving point on the circumference of the larger circle. The line \( BP \) intersects the circumference of the larger circle at another point \( C \). A line passing through point \( P \) and perpendicular to \( BP \) intersects the circumference of the smaller circle at another point \( A \) (if the line \( l \) is tangent to the smaller circle at \( P \), then \( A = P \)).

1. Find the set of values of the expression \( BC^{2} + CA^{2} + AB^{2} \).
2. Find the locus of the midpoint of the segment \( AB \).

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