
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 AB=2a, there is a point C on the semicircle with AB as the diameter. Let the midpoint of AB be O, and $\angle AOC=60°$.
(1) Take a point P on $\widehat{BC}$. If $\angle BOP=2\theta$, express PA+PB+PC as a function of $\theta$;
(2) Let $f(\theta)=PA+PB+PC$. For what value of $\theta$ does $f(\theta)$ reach its maximum, and what is the maximum value?

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