
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 $xOy$, there is a curve $C_{1}: x+y=4$, and another curve $C_{2}$ defined by the parametric equations $\begin{cases} x=1+\cos \theta, \\ y=\sin \theta \end{cases}$ (with $\theta$ as the parameter). A polar coordinate system is established with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

$(1)$ Find the polar equations of curves $C_{1}$ and $C_{2}$.

$(2)$ If a ray $l: \theta=\alpha (\rho > 0)$ intersects $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively, find the maximum value of $\dfrac{|OB|}{|OA|}$.

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