
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
Reading material:<br/>Material $1$: For a one-variable quadratic equation about $x$, $ax^{2}+bx+c=0$ $(a\neq 0)$, the two real roots $x_{1}$ and $x_{2}$ and the coefficients $a$, $b$, $c$ have the following relationship: $x_{1}+x_{2}=-\frac{b}{a}$, $x_{1}x_{2}=\frac{c}{a}$.<br/>Material $2$: Given that the two real roots of the one-variable quadratic equation $x^{2}-x-1=0$ are $m$ and $n$, find the value of $m^{2}n+mn^{2}$.<br/>Solution: Since $m$ and $n$ are the two real roots of the one-variable quadratic equation $x^{2}-x-1=0$, therefore $m+n=1$, $mn=-1$. Then $m^{2}n+mn^{2}=mn(m+n)=-1\times 1=-1$. Based on the above materials and your knowledge, complete the following questions:<br/>$(1)$ Application: If the two real roots of the one-variable quadratic equation $2x^{2}+3x-1=0$ are $x_{1}$ and $x_{2}$, then $x_{1}+x_{2}=$______, $x_{1}x_{2}=\_\_\_\_\_\_$.<br/>$(2)$ Analogy: Given that the two real roots of the one-variable quadratic equation $2x^{2}+3x-1=0$ are $m$ and $n$, find the value of $m^{2}+n^{2}$;<br/>$(3)$ Enhancement: Given real numbers $s$ and $t$ satisfy $2s^{2}+3s-1=0$, $2t^{2}+3t-1=0$ and $s\neq t$, find the value of $\frac{1}{s}-\frac{1}{t}$.

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