
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
Let $\triangle ABC$ have vertices $A$ at the origin $(0,0)$, $B$ at $(8,0)$, and $C$ at $(1,7)$. Suppose you guess that the point $P = (3,4)$ might be a point of interest (e.g., aiming to minimize some combined distance property). Calculate the sum of distances from $P$ to $A$, $B$, and $C$ and express it as $m\sqrt{p} + n\sqrt{q}$, where $m, n, p, q$ are integers. What is $m+n+p+q$?

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