
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 that the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=1+\frac{1}{2}t}\\{y=\sqrt{3}+\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter), establish a polar coordinate system with the origin as the pole and the non-negative x-axis as the polar axis. The polar coordinate equation of curve $C$ is $\rho =4\sin \theta$.  
$(1)$ Find the rectangular coordinate equation of curve $C$ and the polar coordinate equation of line $l$.  
$(2)$ If $M(1,\sqrt{3})$, and the line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac{{|MB|}}{{|MA|}}+\frac{{|MA|}}{{|MB|}}$.

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