
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 parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=4-\frac{{\sqrt{2}}}{2}t}\\{y=4+\frac{{\sqrt{2}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis. The polar coordinate equation of curve $C$ is $\rho =8\sin \theta $, and $A$ is a point on curve $C$.  
$(1)$ Find the maximum distance from $A$ to the line $l$;  
$(2)$ If point $B$ is the intersection point of line $l$ and curve $C$ in the first quadrant, and $∠AOB=\frac{{7π}}{{12}}$, find the area of $\triangle AOB$.

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