
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
In triangle $ABC$, points $G$, $H$, and $I$ are chosen on sides $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, so that $BG:GC = CH:HA = AI:IB = 2:1.$ Line segments $\overline{AG},$ $\overline{BH},$ and $\overline{CI}$ intersect at points $X$, $Y$, and $Z$. Compute $\frac{[XYZ]}{[ABC]}.$

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