
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 equation of line $l$ is $\begin{cases}x=t\cos \alpha \\ y=2+t\sin \alpha\end{cases} (t$ is a parameter, $0\leqslant \alpha < \pi)$, and the parametric equation of curve $C$ is $\begin{cases}x=2\cos \beta \\ y=2+2\sin \beta\end{cases} (\beta$ is a parameter$)$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis to establish a polar coordinate system.  
$(1)$ Find the polar equation of curve $C$;  
$(2)$ Suppose $C$ intersects $l$ at points $M$, $N$ (different from the origin), find the maximum value of $|OM|+|ON|$.

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