
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 $x$ $\left\{\begin{array}{l}{\frac{x-1}{2}≤\frac{2x+3}{6}}\\{x+1>a+3}\end{array}\right.$ has a solution, and the fractional equation about $y$ $\frac{a+1}{y-2}+\frac{3}{2-y}=2$ has a non-negative solution, then the sum of all integers $a$ that satisfy the conditions is ____.

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