
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
Consider a rectangle \(ABCD\) with \(AB = 2\) and \(BC = 1\). Points \(E\) and \(F\) are on \(\overline{BC}\) and \(\overline{CD}\), respectively, such that \(\triangle AEF\) is equilateral. A rectangle \(A'B'C'D'\) with longer side parallel to \(ABCD\) and one vertex at \(B\) is inscribed such that \(AB' = x\), the shorter side of rectangle \(A'B'C'D'\). Find \(x\) in terms of simplified radical form. Assume \(AD\) is the vertical side.

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