
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 rectangular coordinate system $xOy$, curve $C_1$: $\frac{{x}^{2}}{4}+{y}^{2}=1$, curve $C_2$: $\left\{\begin{array}{l}x=2+2cosϕ \\ y=2sinϕ\end{array}\right.$ (where $ϕ$ is a parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis.
1. Find the polar equations of curves $C_1$ and $C_2$;
2. The polar equation of ray $l$ is $θ=α$ (where $ρ≥0$). If $l$ intersects with $C_1$ and $C_2$ at points $A$ and $B$, respectively, which are distinct from the pole, find the maximum value of $\frac{|OB|}{|OA|}$.

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