
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 root, then such an equation is called a "double root equation". The correct statements about double root equations are____ (just fill in the numbers).  
① The equation $x^{2}+2x-8=0$ is a double root equation;  
② The equation $x^{2}-3x+2=0$ is a double root equation;  
③ If $\left(x-2\right)\left(mx+n\right)=0$ is a double root equation, then $4m^{2}+5mn+n^{2}=0$;  
④ If the point $\left(p,q\right)$ lies on the graph of the inverse proportion function $y=\frac{2}{x}$, then the equation in terms of $x$, $px^{2}+3x+q=0$, is a double root equation.

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