
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
A point  $P$  is chosen uniformly at random in the interior of triangle  $ABC$  with side  lengths  $AB = 5$ ,  $BC = 12$ ,  $CA = 13$ . The probability that a circle with radius  $\frac13$  centered at  $P$  does not intersect the perimeter of  $ABC$  can be written as  $\frac{m}{n}$  where  $m, n$  are relatively prime positive integers. Find  $m + n$ .

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