
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 that $P$ is a point inside or on the boundary of a convex polygon with vertices $A_1, A_2, \ldots, A_n$, find the point $P$ such that $f(P)=\sum_{i=1}^{n}\left|P-A_{i}\right|$ is maximized.

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