
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 there exists a solution for the system of linear inequalities in one variable $x$: $\left\{\begin{array}{l}{\frac{3x-1}{2} \geq x+1}\\{2x-4 < a}\end{array}\right.$, and the solution for the fractional equation in variable $y$: $\frac{ay-3}{y-1}-2=\frac{y-5}{1-y}$ is an integer, then the sum of all integers $a$ that satisfy the conditions is ____.

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