
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 $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The polar equation of curve $C_{1}$ is $\rho=4\cos \theta$, and the polar equation of line $l$ is $\rho\cos (\theta+ \frac {\pi}{4})=2 \sqrt {2}$. The two curves intersect at points $A$ and $B$.  
$(1)$ Find the polar coordinates of points $A$ and $B$;  
$(2)$ Let $P$ be a moving point on curve $C_{2}$: $\begin{cases} x=2\cos \phi \\ y=\sin \phi \end{cases}$ (where $\phi$ is a parameter), find the minimum value of the area of $\triangle PAB$.

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