
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 a quadratic equation in $x$, $ax^{2}+bx+c=0$ where $a\neq 0$ has two real roots. By completing the square of the equation, it is found that ${x}_{1}+{x}_{2}=-\frac{b}{a}$ and ${x}_{1}⋅{x}_{2}=\frac{c}{a}$. This relationship between the roots and coefficients was first discovered by Vieta, and is known as Vieta's theorem. Based on the above conclusion, answer the following questions:<br/>Given that $x_{1}$ and $x_{2}$ are the two real roots of the quadratic equation $2x^{2}-3x-1=0$, find the values of the following expressions:<br/>$(1)\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}$;<br/>$(2){({x}_{1}^{2}-{x}_{2}^{2})}^{2}$;<br/>$(3)2{x}_{1}^{2}+3{x}_{2}$.

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