
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 curve $C_{1}: x+y=1$ and the curve $C_{2}$: 
$$
\begin{cases} 
x=2+2\cos \phi \\
y=2\sin \phi 
\end{cases}
$$
with $\phi$ as the parameter, $\phi \in [0,2\pi)$. Establish a polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

$(I)$ Write the polar equations of curves $C_{1}$ and $C_{2}$;

$(II)$ In the polar coordinate system, it is known that point $A$ is the common point of the ray $l: \theta=\alpha$ ($\rho \geqslant 0$) and $C_{1}$, and point $B$ is the common point of $l$ and $C_{2}$. When $\alpha$ varies in the interval $\left[0, \frac{\pi}{2}\right]$, find the maximum value of $\frac{|OB|}{|OA|}$.

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