
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 in the plane with  $AB = 13$ ,  $BC = 14$ ,  $AC = 15$ . Let  $M_n$  denote the smallest possible value of  $(AP^n + BP^n + CP^n)^{\frac{1}{n}}$  over all points  $P$  in the plane. Find  $\lim_{n \to \infty} M_n$ .

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