
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
Let $ABC$ be a triangle inscribed in a given circle $k$, and let $P$ be any point in the plane of $ABC$. Denote by $m$ the minimum of the sum $AP + BP + CP$. Among the triangles inscribed in circle $k$, for which triangle will $m$ be maximal?

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