
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 $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. It is known that curve $C$: $\rho \sin^2\theta -4\cos \theta =0$, and line $l$ passes through point $M(0,4)$ with a slope of $-2$.

$(1)$ Convert the polar equation of curve $C$ into a Cartesian coordinate equation, and write the standard parametric equation of line $l$;

$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $|AB|$.

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