
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
Conrad is trying to determine the Fermat point $P$ of $\triangle DEF$, where $D$ is at the origin, $E$ is at $(8,0)$, and $F$ is at $(5,7)$. He estimates that the point is at $P = (5,3)$. Compute the sum of the distances from $P$ to the vertices of $\triangle DEF$. If the sum is found to be $p\sqrt{a} + q\sqrt{b}$, where $p$, $q$, $a$, and $b$ are integers, find $p+q$.

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