
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
For the interval $\left[a,b\right]\left(a \lt b\right)$, if the function $y=f\left(x\right)$ satisfies the following conditions simultaneously: ① $f\left(x\right)$ is a monotonic function on $\left[a,b\right]$; ② the range of the function $y=f\left(x\right)$ for $x\in \left[a,b\right]$ is $\left[a,b\right]$, then the interval $\left[a,b\right]$ is called the "maintaining value" interval of the function $f\left(x\right)$.

(1) Find all "maintaining value" intervals of the function $y=x^{2}$;

(2) Does the function $y=x^{2}+m\left(m\neq 0\right)$ have a "maintaining value" interval? If it exists, find the range of $m$; if it does not exist, explain the reason.

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