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

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