
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
In triangle  $ABC$ ,  $BC=3, CA=4$ , and  $AB=5$ . For any point  $P$  in the same plane as  $ABC$ , define  $f(P)$  as the sum of the distances from  $P$  to lines  $AB, BC$ , and  $CA$ . The area of the locus of  $P$  where  $f(P)\leq 12$  is  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

*Proposed by Yannick Yao*

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