
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 the square \( ABCD \), and the point of tangency cuts off a segment of 2 cm from vertex \( A \). Two tangents are drawn from point \( C \) to this circle. Find the side of the square if the angle between the tangents is \( 30^\circ \), and it is known that \( \sin 15^\circ = \frac{\sqrt{3}-1}{2 \sqrt{2}} \).

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