
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 $\left\{{\begin{array}{l}{x=2+\frac{1}{2}t}\\{y=\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of curve $C$ is $\rho \sin ^{2}\theta -4\cos \theta =0$.  
$(1)$ Find the general equation of line $l$ and the Cartesian equation of curve $C.  
$(2)$ Given that line $l$ intersects curve $C$ at points $A$ and $B$, and $M(2,0)$, find the value of $|MA||MB|$.

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