
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
Consider triangle $ABC$ where $A$ is positioned at the origin $(0,0)$, $B$ at $(12,0)$, and $C$ at $(4,7)$. Linda considers a pont $P = (5,3)$ and tries to find the sum of the distances from $P$ to the vertices of the triangle. If she finds the sum as $a\sqrt{b} + c\sqrt{d}$, where $a$, $b$, $c$, and $d$ are integers, what is $a + b + c + d$?

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