
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 $(xOy)$, for line $l$ described by $\begin{cases}x=2+t,\\y=2-t\end{cases}$ (where $t$ is a parameter), taking the origin as the pole and the positive $x$-axis as the polar axis, let's establish a polar coordinate system. For curve $C$: $\rho=2\sin\theta$,
$(1)$ find the polar equation of line $l$ and the cartesian equation of curve $C$;
$(2)$ let ray $\theta=\alpha$ (where $\rho\geqslant 0, 0 < \alpha < \frac{\pi}{2}$) intersect line $l$ and curve $C$ at points $M$ and $N$ (different from point $O$) respectively, find the maximum value of $\frac{|ON|}{|OM|}$.

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