
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  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .

*Proposed by Michael Ren*

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