
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 $XYZ$, points $G$, $H$, and $I$ lie on sides $\overline{YZ}$, $\overline{XZ}$, and $\overline{XY}$, respectively, such that $YG:GZ = XH:HZ = XI:IY = 2:1$. Line segments $\overline{XG}$, $\overline{YH}$, and $\overline{ZI}$ intersect at points $P$, $Q$, and $R$, respectively. Compute the ratio $\frac{[PQR]}{[XYZ]}$.

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