
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 equation of line $l$ is $x=4$, and the parametric equation of curve $C$ is $\begin{cases} x=1+\sqrt{2}\cos\varphi \\ y=1+\sqrt{2}\sin\varphi \end{cases}$ (where $\varphi$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis.  
$(1)$ Find the polar equations of line $l$ and curve $C$;  
$(2)$ If the ray $\theta=\alpha(\rho > 0, 0 < \alpha < \frac{\pi}{4})$ intersects curve $C$ at points $O, A$ and intersects line $l$ at point $B$, find the range of values for $\frac{|OA|}{|OB|}$.

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