
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 $r$ ($R > r$). 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 $B P$ intersects the circumference of the larger circle at another point $C$. A line $l$ passing through point $P$ and perpendicular to $B P$ intersects the circumference of the smaller circle at another point $A$ (if $l$ is tangent to the smaller circle at $P$, then $A = P$).

(1) Find the set of values for the expression $BC^{2} + CA^{2} + AB^{2}$;
(2) Determine the locus of the midpoint of the segment $A B$.

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