
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 $xOy$, curve $C_{1}$ is given by the parametric equations $\begin{cases} x=\cos t \\ y=1+\sin t \end{cases}$ (with $t$ as the parameter), and curve $C_{2}$ is given by the Cartesian equation $x^{2}+(y-2)^{2}=4$. Establishing a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of ray $l$ is $\theta=\alpha$ $(0 < \alpha < \pi)$.

$(1)$ Find the polar equations of curves $C_{1}$ and $C_{2}$;  
$(2)$ Let points $A$ and $B$ be the intersections of ray $l$ with curves $C_{1}$ and $C_{2}$, excluding the origin. Find the maximum value of $|AB|$.

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