
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
Given the system of inequalities $\left\{{\begin{array}{l}{3(2x-1)<2x+8①}\\{2+\frac{{3(x+1)}}{8}>3-\frac{{x-1}}{4}②}\end{array}}\right.$.<br/>$(1)$ Find the solution set of this system of inequalities and write down its integer solutions;<br/>$(2)$ If the integer solutions above satisfy the inequality $ax+6\leqslant x-2a$, simplify $|a+1|-|a-1|$.

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