
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
Given an interval $[a, b]$ where $a < b$, if a function $y = f(x)$ satisfies the following conditions simultaneously: 1) $f(x)$ is a monotonic function on $[a, b]$; 2) the range of the function $y = f(x)$, $x \in [a, b]$ is $[a, b]$, then the interval $[a, b]$ is called the "value-preserving" interval of the function $f(x)$.

1. Find all "value-preserving" intervals for the function $y = x^2$.
2. Does the function $y = x^2 + m$ ($m \neq 0$) have a "value-preserving" interval? If it exists, find the range of $m$; if not, explain the reason.

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