
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
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is 
$$
\begin{cases} 
x=3\cos \theta, \\ 
y=\sin \theta, 
\end{cases}
$$
where $\theta$ is the parameter. The parametric equation of line $l$ is 
$$
\begin{cases}
x=a+4t, \\ 
y=1-t,
\end{cases}
$$
where $t$ is the parameter.

$(1)$ If $a=-1$, find the coordinates of the intersection points of $C$ and $l$.

$(2)$ If $a=8$, find the maximum distance from a point on $C$ to $l$.

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