
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
Two different points, $E$ and $F$, lie on opposite sides of line $AB$ such that triangles $ABE$ and $ABF$ are congruent with $AB = 12$, $BE = AF = 13$, and $EA = FB = 20$. Another point $G$ lies on line segment $AB$ such that $AG = 5$ and $GB = 7$. Determine the area of intersection of triangles $ABE$ and $ABG$.

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