
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 equation about $y$ $\frac{{a-({1-y})}}{3}=y-2$ has a non-negative integer solution, and the solution set of the inequality system about $x$ $\left\{{\begin{array}{l}{\frac{{x-a}}{2}≥2}\\{x-4≤3({x-2})}\end{array}\right.$ is $x\geqslant 1$, determine the sum of all integers $a$ that satisfy the conditions.

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