
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 $ABC$ with $AB = BC = x$, $\overline{BD}$ is an altitude. Extend $\overline{AC}$ to point $F$ such that $\overline{CF} = 15$. Let point $E$ be on $\overline{AF}$ such that $BE = 15$. Given that the values of $\sin \angle CBE$, $\sin \angle DBE$, and $\sin \angle ABE$ form an arithmetic progression, and the values of $\cos \angle DBE$, $\cos \angle CBE$, $\cos \angle DBC$ form a geometric progression, find the area of triangle $ABC$.

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