
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 solution set of the inequality system about $x$ is $\left\{{\begin{array}{l}{3x-a＞2({1-x})}\\{\frac{{x-1}}{2}≥\frac{{x+2}}{3}-1}\end{array}}\right.$ is $x\geqslant 1$, and the fractional equation about $y$ $\frac{y}{{y+1}}+\frac{a}{{y-1}}=1$ has an integer solution, determine the number of integers $a$ that satisfy the conditions.

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