
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 $x0y$, we have curve $C_1$: $x+y=2$ and curve $C_2$ parameterized by $\begin{cases} x=3+3\cos \phi \\ y=3\sin \phi \end{cases}$ with parameter $\phi \in [0,2\pi)$. Adopting the 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 curves $C_1$ and $C_2$;
$(2)$ In the polar coordinate system, it is known that the ray $l$: $\theta=a$ (with $\rho \geqslant 0$) intersects $C_1$ at point $A$ and $C_2$ at point $B$ (distinct from point $O$). When $α$ varies in the interval $\left[0, \frac{\pi}{4}\right]$, find the range of the ratio $\frac{|OB|}{|OA|}$.

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