
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 a circle $O: x^2 + y^2 = 4$, and a 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 $P(x_0, y_0)$ ($x_0 \neq \pm1$) 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 the lines $AP_1$ and $AP_2$ intersect the y-axis at $(0, m)$ and $(0, n)$ respectively. Is $m \cdot n$ a constant value? If so, find the constant value; if not, explain why.

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