
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  $ABC$  be a triangle with  $AB = 42$ ,  $AC = 39$ ,  $BC = 45$ . Let  $E$ ,  $F$  be on the sides  $\overline{AC}$  and  $\overline{AB}$  such that  $AF = 21, AE = 13$ . Let  $\overline{CF}$  and  $\overline{BE}$  intersect at  $P$ , and let ray  $AP$  meet  $\overline{BC}$  at  $D$ .  Let  $O$  denote the circumcenter of  $\triangle DEF$ , and  $R$  its circumradius.  Compute  $CO^2-R^2$ .

*Proposed by Yang Liu*

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