
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 inscribed in circle  $\Gamma$ , centered at  $O$  with radius  $333.$  Let  $M$  be the midpoint of  $AB$ ,  $N$  be the midpoint of  $AC$ , and  $D$  be the point where line  $AO$  intersects  $BC$ . Given that lines  $MN$  and  $BO$  concur on  $\Gamma$  and that  $BC = 665$ , find the length of segment  $AD$ .

*Author: Alex Zhu*

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