
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 the parametric equations of the line $l$ as $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\sqrt{3}+\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is the parameter), curve $C$ is a circle with center at point $C(0,2)$ passing through the origin. Establish a polar coordinate system with the origin as the pole and the non-negative x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equation of the line $l$ and curve $C$;<br/>$(2)$ 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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