
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 equation of line $l$ is $x+my-2m-1=0$, where $m\in \mathbb{R}$ and $m\neq 0$.
$(1)$ If the sum of the intercepts of line $l$ on the $x$-axis and $y$-axis is $6$, find the value of the real number $m$;
$(2)$ Suppose line $l$ intersects the positive half of the $x$-axis and $y$-axis at points $A$ and $B$ respectively, and $O$ is the origin. Find the equation of line $l$ when the area of $\triangle AOB$ is minimized.

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