
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 $l: kx-y+2+k=0$ where $k\in \mathbb{R}$.  
$(1)$ If the line does not pass through the fourth quadrant, find the range of $k$.  
$(2)$ If the line $l$ intersects the negative half-axis of the $x$-axis at point $A$ and the positive half-axis of the $y$-axis at point $B$, and the area of $\triangle AOB$ is $S$ (where $O$ is the origin), find the minimum value of $S$ and the equation of line $l$ at that time.

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