
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
In triangle \( ABC \), \( M \) is the midpoint of \( \overline{BC} \), \( AB = 15 \), and \( AC = 20 \). Let \( E \) be on \( \overline{AC} \), and \( F \) be on \( \overline{AB} \), and let \( G \) be the intersection of \( \overline{EF} \) and \( \overline{BM} \). If \( AE = 3AF \), then find \( \frac{EG}{GF} \).

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