
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
We construct three circles:  $O$  with diameter  $AB$  and area  $12+2x$ ,  $P$  with diameter  $AC$  and area  $24+x$ , and  $Q$  with diameter  $BC$  and area  $108-x$ .  Given that  $C$  is on circle  $O$ , compute  $x$ .

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