
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
1. In triangle \( \triangle ABC \), \( AB = 12 \), \( AC = 16 \), \( M \) is the midpoint of \( BC \). Points \( E \) and \( F \) are on sides \( AC \) and \( AB \) respectively. The line \( EF \) intersects \( AM \) at \( G \). If \( AE = 2AF \), find the ratio \(\frac{EG}{GF}\).
2. \( E \) is a point outside a circle. The chords \( EAB \) and \( EDC \) form an angle of \( 40^{\circ} \). If \( AB = BC = CD \), find \( \angle ACD \).

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