
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
Solve the inequalities.<br/>$(1)$ Find the positive integer solutions of the inequality $\frac{5(x-1)}{6}-1＜\frac{x+2}{3}$.<br/>$(2)\left\{\begin{array}{l}3x-2≤x+6\\ \frac{5x+3}{2}＞x\end{array}\right.$.

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