
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 equation of curve $C_{1}$ is $x^{2}+y^{2}=1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established with the same unit length. It is known that the polar equation of line $l$ is $\rho(2\cos \theta - \sin \theta) = 6$.  
$(1)$ After extending the x-coordinate of all points on curve $C_{1}$ by $\sqrt {3}$ times and the y-coordinate by $2$ times to obtain curve $C_{2}$, write down the Cartesian equation of line $l$ and the parametric equation of curve $C_{2}$;  
$(2)$ Let $P$ be any point on curve $C_{2}$. Find the maximum distance from point $P$ to line $l$.

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