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

(1) Find the polar equations of curves $(C_{1})$ and $(C_{2})$;
(2) Let $A$ and $B$ be the intersection points 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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