
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 at the point of tangency, it segments an arc starting from vertex \( A \) with a length of \( \sqrt{2} + \sqrt{2 - \sqrt{2}} \) cm. From point \( C \), two tangents are drawn to this circle. Find the side of the square, given that the angle between the tangents is \( 45^\circ \), and it is known that \( \sin 22.5^\circ = \frac{\sqrt{2 - \sqrt{2}}}{2} \).

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