
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
Definition: The equations $ax-b=0$ and $bx-a=0$ (where $a$ and $b$ are constants not equal to zero) with respect to $x$ are called "opposite equations." For example, the equations $2x-1=0$ and $x-2=0$ are opposite equations.  
$(1)$ If the equations $4x-3=0$ and $3x-c=0$ are opposite equations, then $c=\_\_\_\_\_\_$.  
$(2)$ If the equations $4x+3m+1=0$ and $5x-n+2=0$ are opposite equations, find the value of $m\div n$.  
$(3)$ If the equation $3x-c=0$ and its opposite equation both have integer solutions, find the value of the integer $c$.

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