
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
Consider a regular octagon $ABCDEFGH$. By connecting alternate vertices of the octagon, we create a larger equilateral triangle $\triangle ACE$, and by connecting adjacent vertices starting from A, B, and an appropriate midpoint, we form $\triangle ABM$. Compute the ratio $[\triangle ABM]/[\triangle ACE]$, where $M$ is the midpoint of the line segment $\overline{BC}$.

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