
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 linear inequalities in one variable $x$ $\left\{{\begin{array}{l}{\frac{{x+3}}{2}≤4}\\{2x-a≥2}\end{array}}\right.$ has at least $2$ integer solutions, and the fractional equation in variable $y$ $\frac{{a-1}}{{y-2}}+\frac{4}{{2-y}}=2$ has a non-negative integer solution, then the sum of all integer values of $a$ that satisfy the conditions is ______.

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