
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
A semicircle has diameter  $\overline{AD}$  with  $AD = 30$ . Points  $B$  and  $C$  lie on  $\overline{AD}$ , and points  $E$  and  $F$  lie on the arc of the semicircle. The two right triangles  $\vartriangle BCF$  and  $\vartriangle CDE$  are congruent. The area of  $\vartriangle  BCF$  is  $m\sqrt{n}$ , where  $m$  and  $n$  are positive integers, and  $n$  is not divisible by the square of any prime. Find  $m + n$ .
![Image](https://cdn.artofproblemsolving.com/attachments/b/c/c10258e2e15cab74abafbac5ff50b1d0fd42e6.png)

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