
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
Given the system of linear inequalities in one variable x: $\left\{{\begin{array}{l}{x-1≥\frac{{4x-1}}{3},}\\{5x-1＜a}\end{array}}\right.$, and the fractional equation in variable y: $\frac{{y-1}}{{y+1}}=\frac{a}{{y+1}}-2$, if the solution set of the system of linear inequalities is x≤-2 and the solution of the fractional equation is a negative integer, calculate the sum of all integers a that satisfy the conditions.

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