
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 that $\left(2,m\right)$ is the direction vector of the line $l:x+y+1=0$, $M\left(1,0\right)$, $N\left(2m-1,m\right)$, and $A$ is the midpoint of $MN$.  
$(1)$ Find the coordinates of point $A$.  
$(2)$ If the line $l'$ passes through point $A$ and the intercept on the $y$-axis is $\frac{1}{2}$ of the intercept on the $x$-axis, find the equation of the line $l'$.

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