
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
Triangle $ABC$ has $AB = 15, BC = 13$, and $AC = 14$. The points $D, E$, and $F$ are the feet of the perpendiculars from $A$, $B$, and $C$ to the sides $BC$, $CA$, and $AB$ respectively. Let $G$ be the centroid of triangle $ABC$. Find $XA + XB + XC$.

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