
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 $l: (2m+1)x - (3+m)y + m - 7 = 0$.  
$(1)$ For what value of $m$ is the distance from point $Q(3,4)$ to line $l$ the greatest? Find the maximum value.  
$(2)$ If line $l$ intersects the negative half-axis of the $x$-axis and $y$-axis at points $A$ and $B$ respectively, find the minimum area of $\triangle AOB$ (where $O$ is the origin) and the equation of line $l$ at this point.

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