
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
In the Cartesian coordinate system, given points $A(3,2m-1)$ and $B(n+1,-1)$ are not coincident.<br/>$(1)$ If $AB$ <span class="special-symbol">∥</span> $y$-axis, find the value of $n$;<br/>$(2)$ If point $A$ is moved up by $2$ units and then left by $3$ units to get point $B$, find the values of $m$ and $n$.

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