
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 that the center of circle $M$ lies on the line $y=\frac{1}{2}x$, circle $M$ is tangent to the $y$-axis, and the chord intercepted by circle $M$ on the positive $x$-axis has a length of $2\sqrt{3}$.  
$(1)$ Find the standard equation of circle $M$;  
$(2)$ If the line $l$ passing through point $Q(2,2)$ with slope $k$ intersects circle $M$ at points $A$ and $B$, and point $P(2,-1)$, and the area of $\triangle PAB$ is $3\sqrt{3}$, find the equation of line $l$.

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