
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
Elective 4-4: Coordinate System and Parametric Equations

Given the line $l: x+y-3=0$, and the curve $C: \begin{cases}x=2\cos \theta+2 \\ y=2\sin \theta\end{cases} \left(\theta \text{ is the parameter}\right)$. Establish a polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

$(1)$ Find the polar equations of line $l$ and curve $C$ respectively;

$(2)$ If the ray $m: \theta =\alpha \left(-\dfrac{\pi }{4} < \alpha < \dfrac{\pi }{2}, \rho \geqslant 0\right)$ intersects line $l$ and curve $C$ at points $M$ and $N$ respectively ($N$ is different from the origin $O$), find the maximum value of $\dfrac{|ON|}{|OM|}$.

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