
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 the system of equations in terms of $x$ and $y$: $\left\{\begin{array}{l}{x+2y-6=0}\\{x-2y+mx+5=0}\end{array}\right.$  
$(1)$ Please directly write down all positive integer solutions of the equation $x+2y-6=0$;  
$(2)$ If the solution of the system satisfies $x+y=0$, find the value of $m$;  
$(3)$ Regardless of the value of the real number $m$, the equation $x-2y+mx+5=0$ always has a fixed solution. Please directly write down this solution;  
$(4)$ If the solution of the system has $x$ as an integer and $m$ as an integer, find the value of $m$.

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