
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 fractional equation about $x$: $\frac{x+m}{x+2}-\frac{m}{x-2}=1$ has a solution not exceeding $6$, and the inequality system about $y$: $\left\{\begin{array}{l}{m-6y>2}\\{y-4\leq 3y+4}\end{array}\right.$ has exactly four integer solutions, then the sum of the integers $m$ that satisfy the conditions is ____.

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