
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, vertices A, B, and C are located at (0,0), (12,0), and (5,7) respectively. Jill guesses that the Fermat point $P$ of $\triangle ABC$ might be at $(5,3)$. Calculate the sum of distances from $P$ to each vertex and express the sum as $m\sqrt{p} + n\sqrt{q}$, where $p$ and $q$ are distinct integers and $m$ and $n$ are integers. What is $m + n$?

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