
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 plane rectangular coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate equations of the line $l_{1}$ and the curve $C$ are $\rho \cos \theta =2$ and $\rho =4\sin \theta$ respectively. The polar coordinates of point $P$ are $(4,\frac{π}{4})$. <br/>$(1)$ Find the rectangular coordinate equations of the line $l_{1}$ and the curve $C$; <br/>$(2)$ In the polar coordinate system, given that the ray $l_{2}$: $\theta =\alpha (0 \lt \alpha\ \ \lt \frac{π}{2})$ intersects $l_{1}$ and $C$ at points $A$ and $B$ respectively, and $|OA|⋅|OB|=16+8\sqrt{3}$, find the area of $\triangle POB$.

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