
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
Let right  $\triangle ABC$  have  $AC = 3$ ,  $BC = 4$ , and right angle at  $C$ . Let  $D$  be the projection from  $C$  to  $\overline{AB}$ . Let  $\omega$  be a circle with center  $D$  and radius  $\overline{CD}$ , and let  $E$  be a variable point on the circumference of  $\omega$ . Let  $F$  be the reflection of  $E$  over point  $D$ , and let  $O$  be the center of the circumcircle of  $\triangle ABE$ . Let  $H$  be the intersection of the altitudes of  $\triangle EFO$ . As  $E$  varies, the path of  $H$  traces a region  $\mathcal R$ . The area of  $\mathcal R$  can be written as  $\tfrac{m\pi}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $\sqrt{m}+\sqrt{n}$ .

*Proposed by ApraTrip*

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