
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
If the solution set of the inequality system about $x$ is $\left\{{\begin{array}{l}{\frac{{x+2}}{3}>\frac{x}{2}+1}\\{4x+a<x-1}\end{array}}\right.$ is $x<-2$, and the solution of the fractional equation about $y$ is $\frac{{a+2}}{{y-1}}+\frac{{y+2}}{{1-y}}=2$ is positive, then the sum of all integers $a$ that satisfy the conditions is ______.

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