
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 $AHI$ is equilateral. $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ with $AB = BD = DF = FH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$ if $\overline{AF} = \frac{5}{6}\overline{AH}$.

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