
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
Given a regular tetrahedron \( ABCD \), point \( G \) is the centroid of the base triangle \( \triangle ABC \), and point \( M \) is on segment \( DG \) such that the sine of the obtuse angle \( AMB \) is \(\frac{3 \sqrt{51}}{22}\). Find \(\frac{DM}{MG}\).

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