
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 $l$: $kx-y+1+2k=0$ ($k\in\mathbb{R}$), line $l$ intersects the negative half of the $x$-axis at point $A$ and the positive half of the $y$-axis at point $B$.

$(1)$ Let the area of $\triangle ABO$ be $S$, find the minimum value of $S$ and the equation of line $l$ at this time;

$(2)$ Line $l$ passes through a fixed point $M$, find the minimum value of $|MA|\cdot|MB|$.

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