
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
The solution to the fractional equation $\frac{a-2}{x+3}+1=\frac{3x}{3+x}$ in terms of $x$ is an integer, and the inequality system in terms of $y$ $\left\{\begin{array}{l}{\frac{1}{3}y+1≥\frac{y+3}{2}}\\{\frac{a+y}{2}＜y-1}\end{array}\right.$ has a solution with at most six integer solutions. 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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