
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 \( \triangle ABC \), point \( F \) lies on side \( AC \) such that \( FC = 2AF \). If \( G \) is the midpoint of segment \( BF \) and \( E \) is the intersection point of the line passing through \( A \) and \( G \) with segment \( BC \), calculate the ratio \( \frac{EC}{BE} \).

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