
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
Wendy is trying to locate the Fermat point \( P \) of \(\triangle ABC\), where \( A \) is at the origin, \( B \) is at \( (12,0) \), and \( C \) is at \( (5,8) \). She estimates that the point is at \( P = (5,3) \), and calculates the sum of the distances from \( P \) to the vertices of \(\triangle ABC\). If she obtains \( m\sqrt{a} + n\sqrt{b} \), where \( m \), \( n \), \( a \), and \( b \) are integers, what is \( m + n \)?

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