
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
Consider triangle \(ABC\) where \(A\) is located at the origin \((0,0)\), \(B\) is at \((10,0)\), and \(C\) is at \((7,5)\). An additional point \(D\) is placed on the coordinate plane at \((2,3)\). Evelyn guesses that the Fermat point \(P\) of \(\triangle ABC\) is at \((5,3)\). Compute the sum of distances from \(P\) to \(A\), \(B\), \(C\), and \(D\) and express your result in the form \(m + n\sqrt{p}\), where \(m\), \(n\), and \(p\) are integers. What is \(m + n\)?

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