
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  $AX$  be a diameter of a circle  $\Omega$   with radius  $10$ , and suppose that  $C$  lies on  $\Omega$   so that  $AC = 16$ . Let  $D$  be the other point on  $\Omega$  so  $CX = CD$ . From here, define  $D'$  to be the reflection of  $D$  across the midpoint of  $AC$ , and  $X'$  to be the reflection of  $X$  across the midpoint of  $CD$ . If the area of triangle  $CD'X'$   can be written as  $\frac{p}{q}$  , where  $p, q$  are relatively prime, find  $p + q$ .


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