
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
\(A\) is the center of a semicircle, with radius \(AD\) lying on the base. \(B\) lies on the base between \(A\) and \(D\), and \(E\) is on the circular portion of the semicircle such that \(EBA\) is a right angle. Extend \(EA\) through \(A\) to \(C\), and put \(F\) on line \(CD\) such that \(EBF\) is a line. Now \(EA=1\), \(AC=\sqrt{2}\), \(BF=\frac{2-\sqrt{2}}{4}\), \(CF=\frac{2\sqrt{5}+\sqrt{10}}{4}\), and \(DF=\frac{2\sqrt{5}-\sqrt{10}}{4}\). Find \(DE\).

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