
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 side length 2, and let \(\Gamma\) be a circle with radius \(\frac{1}{2}\) centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on \(\Gamma\), visits all three sides of \(ABC\), and ends somewhere on \(\Gamma\) (not necessarily at the starting point). Express your answer in the form of \(\sqrt{p}-q\), where \(p\) and \(q\) are rational numbers written as reduced fractions.

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