
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
Mason is seeking to ascertain the Fermat point \(P\) of \(\triangle ABC\), where \(A\) is at the origin, \(B\) is at \( (12, 0) \), and \(C\) is at \( (4, 6) \). Thinking that \(P = (5,3)\) could be the Fermat point, he decides to compute the sum of the distances from \(P\) to the vertices. The final expression he achieves is in the form \(x\sqrt{a} + y\sqrt{b}\), where \(x\), \(y\), \(a\), and \(b\) are integers. Compute \(x + y\).

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