
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 extensions of two sides $AB$ and $AD$ of square $ABCD$, and it cuts off a segment of length $6-2\sqrt{5}$ cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side length of the square, given that the angle between the tangents is $36^{\circ}$ and $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$.

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