
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 the following image, there are equilateral triangles $ABC$, $DBE$, $IEF$, and $HIG$. The areas of triangles $DBE$, $IEF$, and $HIG$ are in the ratio $9:16:4$. In what ratio are

1. the lengths of the segments $HI$ and $IE$,
2. the areas of triangles $ABC$ and $HEC$?

(K. Pazourek)

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