
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 the parametric equation of line $l$ as $\begin{cases} x=-1+ \frac{\sqrt{2}}{2}t \\ y= \frac{\sqrt{2}}{2}t \end{cases}$, where $t$ is the parameter, and the curve $C_{1}$: $\rho^{2}\cos^{2}\theta+3\rho^{2}\sin^{2}\theta-3=0$, with the origin as the pole and the positive $x$-axis as the polar axis, establishing a polar coordinate system, the same unit of length is taken in both coordinate systems.  
$(1)$ Find the general equation of line $l$ and the rectangular coordinate equation of curve $C_{1}$;  
$(2)$ Is there a point $P$ on curve $C_{1}$ such that the distance from point $P$ to line $l$ is maximized? If such a point exists, find the maximum distance and the point $P$. If not, please explain why.

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