
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 $C$ with a radius of $2$, and its center is on the positive half of the $x$-axis. The line $3x-4y+4=0$ is tangent to the circle $C$.
$(I)$ Find the equation of circle $C$;
$(II)$ A line $l$ passes through point $Q(0,-3)$ and intersects circle $C$ at two distinct points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If $\overrightarrow{OA} \cdot \overrightarrow{OB}=3$ (where $O$ is the origin), find the equation of line $l$.

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