
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
Dorothy is trying to find the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(12,0)$, and $C$ is at $(4,6)$. She believes that the point $P = (5,3)$ is the Fermat point and computes the sum of the distances from $P$ to the vertices of $\triangle ABC$. Find $AP + BP + CP$ in the form $x\sqrt{a} + y\sqrt{b}$, where $x, y, a, b$ are integers, and determine $x + y$.

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