
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 circle $C$ as $\left\{{\begin{array}{l}{x=5+2\cos\alpha,}\\{y=\sqrt{3}+2\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the origin $O$ as the pole and the positive x-axis as the polar axis to establish a polar coordinate system, the polar equation of line $l_{1}$ is $\rho\cos\theta-\sqrt{3}\rho\sin\theta=3$. Moving line $l_{1}$ 3 units to the left results in line $l_{2}$.  
$(1)$ Find the polar equation of circle $C$ and the Cartesian equation of line $l_{2}$.  
$(2)$ Line $l_{2}$ intersects circle $C$ at points $A$ and $B$. Find the ratio of the length of the major arc $\widehat {AB}$ to the length of the minor arc $\widehat {AB}$.

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