
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
There are three emerging towns located at points A, B, and C, with AB=AC=13 kilometers and BC=10 kilometers. Now, there is a plan to jointly build a central hospital. To facilitate access from all three towns, the hospital needs to be built at point P on the perpendicular bisector of BC. If the sum of the squares of the distances from point P to the three towns is to be minimized, where should point P be located?

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