
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
The square  $BCDE$  is inscribed in circle  $\omega$  with center  $O$ .  Point  $A$  is the reflection of  $O$  over  $B$ . A "hook" is drawn consisting of segment  $AB$  and the major arc  $\widehat{BE}$  of  $\omega$  (passing through  $C$  and  $D$ ). Assume  $BCDE$  has area  $200$ .  To the nearest integer, what is the length of the hook?

*Proposed by Evan Chen*

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