
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
Given the straight line \( y = -\sqrt{3}x + 2\sqrt{3} \), it intersects the \( x \)-axis and \( y \)-axis at points \( B \) and \( C \) respectively. Let point \( A(-2,0) \) and \( P \) be a moving point on the line segment \( BC \). By translating \( BC \) to different positions, find the number of positions for point \( P \) such that \( \angle APO = 30^\circ \).

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