
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 equations for curve $C$ are $\begin{cases}x=3\cos \theta, \\ y=\sin \theta,\end{cases}$ $(\theta \text{ as parameter})$, and the parametric equations for line $l$ are $\begin{cases}x=a+4t, \\ y=1-t,\end{cases}$ $(t \text{ as parameter})$.

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

$(2)$ If the maximum distance from points on $C$ to line $l$ is $\sqrt{17}$, find $a$.

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