
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 $ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE = 14$. The values of $\tan \angle CBE$, $\tan \angle ABE$, and $\tan \angle DBE$ form a geometric series, and the values of $\cot \angle CBE$, $\cot \angle DBC$, $\cot \angle DBE$ form an arithmetic progression. What is the area of triangle $ABC$?

[asy]
pair A,B,C,D,E;
A=(0,0);
B=(4,8);
C=(8,0);
E=(14,0);
D=(4,0);
draw(A--B--E--cycle,linewidth(0.7));
draw(C--B--D,linewidth(0.7));
label("$B$",B,N);
label("$A$",A,S);
label("$D$",D,S);
label("$C$",C,S);
label("$E$",E,S);
[/asy]

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