
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
Let \( \triangle ABC \) be an acute triangle, with \( M \) being the midpoint of \( \overline{BC} \), such that \( AM = BC \). Let \( D \) and \( E \) be the intersection of the internal angle bisectors of \( \angle AMB \) and \( \angle AMC \) with \( AB \) and \( AC \), respectively. Find the ratio of the area of \( \triangle DME \) to the area of \( \triangle ABC \).

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