
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  $\triangle ABC$  satisfies  $AB = 13$ ,  $BC = 14$ , and  $AC = 15$ . Inside  $\triangle ABC$  are three points  $X$ ,  $Y$ , and  $Z$  such that:



- $Y$  is the centroid of  $\triangle ABX$
- $Z$  is the centroid of  $\triangle BCY$
- $X$  is the centroid of  $\triangle CAZ$


    What is the area of  $\triangle XYZ$ ?

*Proposed by Adam Bertelli*

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