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

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