
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 equation $lx-y+1+2k=0$ ($k \in \mathbb{R}$).  
If the line does not pass through the fourth quadrant, find the range of values for $k$;  
The line $l$ intersects the negative half of the x-axis at point A and the positive half of the y-axis at point B, with O being the origin. Let $\triangle AOB$ have an area $S$. 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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