
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 touches the extensions of two sides $AB$ and $AD$ of a square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From the point $C$, two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ}=\frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$.

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