
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 $x$, $ax^{2}+bx+c=0 (a\neq 0)$, has two real roots, where one root is twice the other root, then such an equation is called a "double root equation". For example, the quadratic equation $x^{2}-6x+8=0$ has two roots $x_{1}=2$ and $x_{2}=4$, so the equation $x^{2}-6x+8=0$ is a "double root equation".<br/>$(1)$ By calculation, determine whether $x^{2}-3x+2=0$ is a "double root equation";<br/>$(2)$ If the equation in $x$, $(x-2)(x-m)=0$, is a "double root equation", find the value of the algebraic expression $m^{2}+2m+2$;<br/>$(3)$ Given that the quadratic equation in $x$, $x^{2}-(m-1)x+32=0 (m$ is a constant$)$ is a "double root equation", please directly write down the value of $m$.

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