
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 $y$ is $\left\{{\begin{array}{l}{y-1≥\frac{{2y-1}}{3}}\\{-\frac{1}{2}({y-a})＞0}\end{array}}\right.$ has no solution, and the solution to the fractional equation about $x$ is $\frac{a}{{x+1}}+1=\frac{{x+a}}{{x-1}}$ is negative, then the sum of all integer values of $a$ that satisfy the conditions is ____.

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