
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
Definition: If $x_{1}$ and $x_{2}$ are two integer roots of the equation $ax^{2}+bx+c=0\left(a\neq 0\right)$, and satisfy $|x_{1}-x_{2}|=1$, then this type of equation is called a "natural equation". For example, $\left(x-2\right)\left(x-3\right)=0$ is a "natural equation".<br/>$(1)$ The following equations are "natural equations"______$; (Fill in the numbers)$<br/>①$x^2-\frac{1}{4}=0$;<br/>②$x^{2}+x=1$;<br/>③$x^{2}+3x+2=0$;<br/>$(2)$ If the equation $x^{2}-\left(m+1\right)x+m=0$ is a "natural equation", find the value of $m$.

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