
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
Given the line $l_{1}$: $y=\frac{1}{2}x+3$ intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively.  
$(1)$ Find the coordinates of points $A$ and $B$;  
$(2)$ The line $l_{1}$ is shifted $8$ units to the right to obtain the line $l_{2}$. Find the equation of the line $l_{2}$;  
$(3)$ Let point $P$ be the intersection of line $l_{2}$ and the $x$-axis. Find the area of $\triangle PAB$.

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