
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
Read the following material: Find the solution set of the inequality $\left(2x-1\right)\left(x+3\right) \gt 0$.  
Solution: According to the rule "the product of two numbers with the same sign is positive," we have:  
① $\left\{{\begin{array}{l}{2x-1＞0}\\{x+3＞0}\end{array}}\right.$ or ② $\left\{{\begin{array}{l}{2x-1＜0}\\{x+3＜0}\end{array}}\right.$.  
Solving ①, we get $x＞\frac{1}{2}$. Solving ②, we get $x \lt -3$.  
$\therefore$ The solution set of the inequality is $x＞\frac{1}{2}$ or $x \lt -3$.  
Please solve the following problems using the same method:  
$(1)$ Find the solution set of the inequality $\left(2x-3\right)\left(x+1\right) \lt 0$;  
$(2)$ Find the solution set of the inequality $\frac{{4x-1}}{{x+2}}≥0$.

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