
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 the function $y=f(x)$ satisfies the following conditions on the interval $\left[a,b\right]$: ① $f(x)$ is a monotonic function on the interval $\left[a,b\right]$, ② for $x\in \left[a,b\right]$, the range of the function $f(x)$ is $\left[a,b\right]$, then the interval $\left[a,b\right]$ is called the "value-preserving" interval of the function $f(x)$. If the function $f(x)={x}^{2}-\frac{1}{2}x+m$ has a "value-preserving" interval, the range of the real number $m$ is ______.

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