
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 the rectangular coordinate system \(xOy\), given an equilateral triangle \( \triangle ABC\) (vertices are ordered counterclockwise), vertex \( A(-2,2)\), vertex \( B\) is on the \( y \)-axis, and \( CB\) is parallel to the \( x \)-axis; point \( D(2,6)\), \( E(2,2)\), the radius of circle \( \odot O\) is 2. Find the radius of a circle that is externally tangent to \( \odot O\) and also tangent to both line segments \( AB\) and \( DE\) (accurate to \(10^{-4}\)).

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