
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
Chords \( AB \) and \( CD \) intersect at point \( E \) inside a circle. Let \( M \) be an internal point of segment \( BE \).

The tangent at point \( E \) to the circumcircle of triangle \( DEM \) intersects lines \( BC \) and \( AC \) at points \( F \) and \( G \), respectively. Let \( \frac{AM}{AB} = t \). Find \( \frac{BG}{EF} \).

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