
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
Three unit circles \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) in the plane have the property that each circle passes through the centers of the other two. A square \(S\) surrounds the three circles in such a way that each of its four sides is tangent to at least one of \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\). Find the side length of the square \(S\).

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