
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 \( A B C \), the sides \( A B \) and \( B C \) are equal. A line parallel to the base \( A C \) intersects side \( A B \) at point \( D \) and side \( B C \) at point \( E \), such that each segment \( A D, E C \), and \( D E \) is equal to 2. Point \( F \) is the midpoint of segment \( A C \), and point \( G \) is the midpoint of segment \( E C \). A line segment connects points \( F \) and \( G \). It is known that the angle \( \angle G F C \) is \( \beta \). Find the area of triangle \( A B C \).

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