
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 polar coordinate system is established with the coordinate origin as the pole and the positive half-axis of $x$ as the polar axis. The polar equation of curve $C_1$ is $\rho = 4\cos \theta$, and the polar equation of line $l$ is $\rho\cos \left(\theta + \frac{\pi}{4}\right) = 2\sqrt{2}$.
$(1)$ Find the polar coordinates of the intersection points of curve $C_1$ and line $l$.
$(2)$ Suppose point $P$ is on curve $C_2$: $\begin{cases} x=2\cos \phi \\ y=\sin \phi \end{cases}$ (with $\phi$ as a parameter). Let the intersection points of line $l$ and curve $C_1$ be $A$ and $B$. Find the minimum value of the area of triangle $PAB$.

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