
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 of the line $l$ are $\left\{\begin{array}{l}{x=\frac{\sqrt{5}}{5}t,}\\{y=2+\frac{2\sqrt{5}}{5}t}\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $\rho =4\sin \theta -2\cos \theta$. <br/>$(1)$ Find the general equation of the line $l$ and the Cartesian equation of the curve $C$; <br/>$(2)$ If the intersection points of the line $l$ with the $y$-axis are $M$, and with the curve $C$ are $A$ and $B$, find the value of $\frac{|MA||MB|}{{|MA|}^{2}+|MB|^{2}}$.

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