
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
In equilateral triangle  $XYZ$  with side length  $10$ , define points  $A, B$  on  $XY,$  points  $C, D$  on  $YZ,$  and points  $E, F$  on  $ZX$  such that  $ABDE$  and  $ACEF$  are rectangles,  $XA<XB,$   $YC<YD,$  and  $ZE<ZF$ . The area of hexagon  $ABCDEF$  can be written as  $\sqrt{x}$  for some positive integer  $x$ . Find  $x$ .

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