
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 materials:<br/>Material $1$: If the two roots of a quadratic equation $ax^{2}+bx+c=0 (a\neq 0)$ are $x_{1}$ and $x_{2}$, then ${x}_{1},{x}_{2}=-\frac{b}{a},{x}_{1}{x}_{2}=\frac{c}{a}$<br/>Material $2$: Given real numbers $m$ and $n$ satisfying $m^{2}-m-1=0$, $n^{2}-n-1=0$, and $m\neq n$, find the value of $\frac{m}{n}+\frac{n}{m}$.<br/>Solution: According to the question, $m$ and $n$ are two distinct real roots of the equation $x^{2}-x-1=0$. According to Material $1$, we have $m+n=1$, $mn=-1$.<br/>Therefore, $\frac{m}{n}+\frac{n}{m}=\frac{{m}^{2}+{n}^{2}}{mn}=\frac{{(m+n)}^{2}-2mn}{mn}=\frac{1+2}{-1}=-3$.<br/>Based on the above materials, solve the following problems:<br/>$(1)$ If the two roots of the quadratic equation $5x^{2}+10x-1=0$ are $x_{1}$ and $x_{2}$, then $x_{1}+x_{2}=$______, $x_{1}x_{2}=$______;<br/>$(2)$ Given real numbers $m$ and $n$ satisfying $3m^{2}-3m-1=0$, $3n^{2}-3n-1=0$, and $m\neq n$, find the value of $m^{2}n+mn^{2}$;<br/>$(3)$ Given real numbers $p$ and $q$ satisfying $p^{2}=7q-2$, $2q^{2}=7p-1$, and $p\neq 2q$, find the value of $p^{2}+4q^{2}$.

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