
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
Let line $l_{1}$: $2x+y-1=0$, line $l_{2}$: $x-y+2=0$, and line $l_{3}$: $3x+my-6=0$.

$(1)$ If lines $l_{1}$, $l_{2}$, and $l_{3}$ intersect at the same point, find the value of $m$;  
$(2)$ Suppose line $l$ passes through point $M(2,0)$, and the segment intercepted by lines $l_{1}$ and $l_{2}$ is exactly bisected by point $M$, find the equation of line $l$.

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