
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
If a quadratic equation in terms of $x$, $ax^{2}+bx+c=0$, has two real roots, and one root is twice the other, then such an equation is called a "double root equation." The correct statements about double root equations are: __(select the correct numbers)__<br/>① The equation $x^{2}-x-2=0$ is a double root equation;<br/>② If $\left(x-2\right)\left(mx+n\right)=0$ is a double root equation, then $4m^{2}+5mn+n^{2}=0$;<br/>③ If $p$ and $q$ satisfy $pq=2$, then the equation in terms of $x$, $px^{2}+3x+q=0$, is a double root equation;<br/>④ If the equation $ax^{2}+bx+c=0$ is a double root equation, then it must be true that $2b^{2}=9ac$.

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