
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 $M$ of circle $M$ lies on the $x$-axis, with a radius of $2$, and the chord intercepted by line $l: 3x+4y-1=0$ on circle $M$ has a length of $2\sqrt{3}$, and the center $M$ of circle $M$ lies above line $l$.  
$(1)$ Find the equation of circle $M$;  
$(2)$ Let $A(0,t)$, $B(0,t-6)$ $(2\leq t\leq 4)$, if circle $M$ is the inscribed circle of $\triangle ABC$, find the slope of the lines containing sides $AC$ and $BC$ (expressed in terms of $t$);  
$(3)$ Under the conditions of $(2)$, find the maximum value of the area $S$ of $\triangle ABC$ and the corresponding value of $t$.

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