
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 \\(O: x^{2}+y^{2}=4\\), and line \\(l_{1}: \sqrt {3}x+y-2 \sqrt {3}=0\\) intersects circle \\(O\\) at points \\(A\\) and \\(B\\), with point \\(A\\) in the first quadrant.  
\\((1)\\) Find \\(|AB|\\);  
\\((2)\\) Let \\(P(x_{0},y_{0})(x_{0}\neq ±1)\\) be a moving point on circle \\(O\\), the symmetric point of \\(P\\) with respect to the origin is \\(P_{1}\\), and the symmetric point of \\(P\\) with respect to the \\(x\\)-axis is \\(P_{2}\\). If lines \\(AP_{1}\\) and \\(AP_{2}\\) intersect the \\(y\\)-axis at \\((0,m)\\) and \\((0,n)\\) respectively, is \\(m\cdot n\\) a constant? If yes, find the constant; if no, explain why.

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