
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 right triangle $ABC$ with right angle at $C$, $AB = 1$. Point $D$ is on $\overline{BC}$, point $E$ is on $\overline{AC}$, and line segment $\overline{DE}$ is such that $\triangle ADE$ forms an equilateral triangle. A circle with center at $A$ passes through $D$, and a smaller square with sides parallel to $\triangle ADE$ has vertices on this circle such that one vertex is at $D$. Calculate the radius of the circle.

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