
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 an acute triangle with circumcircle  $\Gamma$ . Let the internal angle bisector of  $\angle BAC$  intersect  $BC$  and  $\Gamma$  at  $E$  and  $N$ , respectively. Let  $A'$  be the antipode of  $A$  on  $\Gamma$  and let  $V$  be the point where  $AA'$  intersects  $BC$ . Given that  $EV=6$ ,  $VA'=7$ , and  $A'N=9$ , compute the radius of  $\Gamma$ .

*Proposed by James Lin.*

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