
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 $ABC$, where $A$ is at the origin $(0,0)$, $B$ is now at $(12,0)$, and $C$ remains at $(3,5)$. If Wanda now guesses that the Fermat point $P$ is at $(5,3)$, calculate 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, find $m + n$.

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