
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 $(xOy)$, there are two curves $C_1: x + y = 4$ and $C_2: \begin{cases} x = 1 + \cos \theta \\ y = \sin \theta \end{cases}$ (where $\theta$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.

(I) Find the polar equations of the curves $C_1$ and $C_2$.

(II) If the line $l: \theta = \alpha (\rho > 0)$ intersects $C_1$ and $C_2$ at points $A$ and $B$ respectively, find the maximum value of $\frac{|OB|}{|OA|}$.

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