
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
Read the following materials:<br/>$(1)$ In the Cartesian coordinate plane, for point $P\left(x,y\right)$, if the coordinates of point $Q$ are $\left(ax+y,x+ay\right)$, where $a$ is a constant, then point $Q$ is called the "$a$-level associated point" of point $P$. For example, the $3$-level associated point of point $P\left(1,4\right)$ is $Q\left(3\times 1+4,1+3\times 4\right)$, which is $Q\left(7,13\right)$.<br/>$(2)$ Given that the $\frac{1}{2}$-level associated point of point $A\left(2,6\right)$ is point $B$, find the coordinates of point $B$.<br/>$(3)$ Given that the "$a$-level associated point" of point $P\left(2,-1\right)$ is $\left(9,b\right)$, find the value of $a+b$.<br/>$(4)$ Given that the "$-4$-level associated point" $N$ of point $M\left(m-1,2m\right)$ lies on the coordinate axis, find the coordinates of point $N$.

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