
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 located on sides \(\overline{BC}\), \(\overline{CA}\), and \(\overline{AB}\) respectively, such that \(BG:GC = CH:HA = AI:IB = 2:1\). Line segments \(\overline{AG}\), \(\overline{BH}\), and \(\overline{CI}\) intersect at points \(S\), \(T\), and \(U\) respectively. Compute the ratio \(\frac{[STU]}{[ABC]}\).

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