
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  $AB$  be diameter of a circle  $\omega$  and let  $C$  be a point on  $\omega$ , different from  $A$  and  $B$ . The perpendicular from  $C$  intersects  $AB$  at  $D$  and  $\omega$  at  $E(\neq C)$ . The circle with centre at  $C$  and radius  $CD$  intersects  $\omega$  at  $P$  and  $Q$ . If the perimeter of the triangle  $PEQ$  is  $24$ , find the length of the side  $PQ$ 

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