
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
Liam is trying to find 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 $P$ is at $(3,3)$. He computes the sum of the distances from $P$ to the vertices of $\triangle DEF$. If he gets a result of $p\sqrt{q} + r\sqrt{s}$, where $p$, $q$, $r$, and $s$ are integers, what is $p + r$?

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