
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 the curve $C$ is 
$$
\begin{cases} 
x=1+\sqrt{2}\cos \varphi \\ 
y=1+\sqrt{2}\sin \varphi
\end{cases}
\quad (\varphi \text{ as the parameter}).
$$
Consider the origin as the pole and the positive half-axis of $x$ as the polar axis to establish a polar coordinate system.
(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 point $O$ and line $l$ at point $B$, find the range of $|\frac{OA}{OB}|$.

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