
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
A triangle \( \triangle ABC \) has vertices \( A \) at the origin (0,0), \( B \) at (10, -2), and \( C \) at (7,6). Zoe wants to find the sum of distances from point \( P = (6,3) \) to each vertex of the triangle. Compute the sum of these distances and express it as \( m + n\sqrt{p} \) where \( m, n, \) and \( p \) are integers. Determine \( m + n + p \).

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