
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
Let  $\omega$  be a circle with radius  $1$ . Equilateral triangle  $\vartriangle ABC$  is tangent to  $\omega$   at the midpoint of side  $BC$  and  $\omega$   lies outside   $\vartriangle ABC$ . If line  $AB$  is tangent to  $\omega$  , compute the side length of   $\vartriangle  ABC$ .

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