
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
Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABI$, shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\triangle ACE$, also shown in boldface. Compute the ratio $[\triangle ABI]/[\triangle ACE]$. Assume that $I$ is the center of the octagon and that each side of the octagon has length $a$.

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