
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 2 concentric circle radii  $ R$  and   $ r$  ( $ R > r$ ) with centre  $ O.$  Fix  $ P$  on the small circle and consider the variable chord  $ PA$  of the small circle. Points  $ B$  and  $ C$  lie on the large circle;  $ B,P,C$  are collinear and  $ BC$  is perpendicular to  $ AP.$ 

**i.)** For which values of  $ \angle OPA$  is the sum  $ BC^2 \plus{} CA^2 \plus{} AB^2$  extremal?

**ii.)** What are the possible positions of the midpoints  $ U$  of  $ BA$  and  $ V$  of  $ AC$  as  $ \angle OPA$  varies?

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