
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 inequality system about $x$ is $\left\{\begin{array}{l}{\frac{x+3}{2}≥x-1}\\{3x+6＞a+4}\end{array}\right.$ has exactly $3$ odd solutions, and the solution to the equation about $y$ is $3y+6a=22-y$ is a non-negative integer, then the product of all integers $a$ that satisfy the conditions is ____.

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