
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
$(1)$ Calculate: $\frac{{x}^{2}+x}{{x}^{2}-2x+1}÷(\frac{2}{x-1}-\frac{1}{x})$;  
$(2)$ Solve the inequality system $\left\{\begin{array}{l}{\frac{2x+1}{3}-\frac{5x-1}{2}＜1}\\{5x-1＜3(x+2)}\end{array}\right.$, and write down its positive integer solutions;

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