
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
Stephanie is attempting to locate a point $Q$ for triangle $\triangle XYZ$, where $X$ is at the origin, $Y$ is at $(10,0)$, and $Z$ is at $(5,3)$. She believes that the point $Q = (5,1)$ minimizes the sum of distances from the vertices of $\triangle XYZ$. Calculate the sum of the distances of $Q$ from each vertex. If the sum is expressed as $u \sqrt{a} + v \sqrt{b}$, where $u, v, a, b$ are integers, what is $u + v$?

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