
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 circle is tangent to the coordinate axes, to one leg, and the hypotenuse of a $30^\circ$-$60^\circ$-$90^\circ$ triangle $DEF$ where $DE=2$. Where the circle intersects two axes and the hypotenuse, locate the center $O$ of the circle and calculate its radius.

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