
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
Three non-overlapping points $A$, $B$, $C$ are known on the number line. Points $A$ and $B$ represent numbers on the number line that are opposite to each other. The distance between points $A$ and $B$ is $m$ (point $A$ is to the left of point $B$). The number represented by point $C$ on the number line is $mn-1$, where $m$ and $n$ are integers.<br/>$(1)$ If $m=4$, find the numbers represented by points $A$ and $B$ on the number line;<br/>$(2)$ If the distances from points $A$ and $B$ to point $C$ are equal, find the difference between $2mn+m(\frac{1}{2}n+3)$ and $3(\frac{1}{2}+m)+1$;<br/>$(3)$ If the distances from points $B$ and $C$ to point $A$ are equal, find the value of $n$.

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