
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 $xOy$, consider the curve $C_1$: $\frac{x^2}{2}+y^2=1$, and the curve $C_2$: $\begin{cases} x=\cos\phi \\ y=1+\sin\phi \end{cases}$ (with $\phi$ as the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.  
$(1)$ Find the polar equations of the curves $C_1$ and $C_2$.  
$(2)$ It is known that the ray $l$: $\theta=\alpha$ ($\rho\geqslant 0$) intersects the curves $C_1$ and $C_2$ respectively at points $A$ and $B$ (different from the origin $O$). When $0 < \alpha < \frac{\pi}{4}$, find the range of $|OA|^2+|OB|^2$.

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