
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 $O: x^{2}+y^{2}=1$ intersects the negative half of the $x$-axis at point $A$ and the positive half of the $y$-axis at point $B$  
$(1)$ If the line $l$ passing through point $C\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ cuts a chord of length $\sqrt{3}$ from circle $O$, find the equation of line $l$;  
$(2)$ If there exists a point $P$ on the circle with center $B$ and radius $r$ such that $PA= \sqrt{2}PO$ (where $O$ is the origin), find the range of values for $r$;  
$(3)$ Let $M(x_{1},y_{1})$ and $Q(x_{2},y_{2})$ be two moving points on circle $O$. The symmetric point of $M$ with respect to the origin is $M_{1}$, and the symmetric point of $M$ with respect to the $x$-axis is $M_{2}$. If the lines $QM_{1}$ and $QM_{2}$ intersect the $y$-axis at $(0,m)$ and $(0,n)$ respectively, is $m\cdot n$ a constant value? If yes, find this constant value; if no, please explain why.

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