
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 circle C: x^2 + y^2 = 3, line l: x + 3y - 6 = 0, point P(x_0, y_0) ∈ l, find the range of x_0 such that there exists point Q ∈ C, satisfying ∠OPQ = 60^{\circ}, where O is the coordinate origin.

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