
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 plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\left\{\begin{array}{l}x=2(t-2\sqrt{2})\\ y=t\end{array}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho ^{2}(1+3\sin ^{2}\theta )=4$.  
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;  
$(2)$ If the ray $\theta =\beta$ (where $\beta \in \left(0,\pi \right)$, and $\tan\beta=-\frac{1}{2}$, $\rho \geqslant 0$) intersects the curve $C$ above the $x$-axis at point $M$ and intersects the line $l$ at point $N$, find $|MN|$.

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