
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 system of inequalities about $x$ is $\left\{\begin{array}{l}{\frac{x}{3}+1≤\frac{x+3}{2}}\\{3x-\frac{a-1}{3}＜0}\end{array}\right.$ has exactly $4$ integer solutions, and the positive integer solutions of the fractional equation about $y$ $\frac{9-a}{y-1}+\frac{2}{1-y}=2$, then the sum of all integers $a$ that satisfy the conditions is ______.

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