
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 \( ABC \) be an equilateral triangle with \( AB = 3 \). Circle \( \omega \) with diameter 1 is drawn inside the triangle such that it is tangent to sides \( AB \) and \( AC \). Let \( P \) be a point on \( \omega \) and \( Q \) be a point on segment \( BC \). Find the minimum possible length of the segment \( PQ \).

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