
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 the function $f(x)$ is a quadratic function, and the function $g(x) = x^2 - 2$, $f(x) + g(x)$ is an odd function. Moreover, the equation $f(x) = 3x + 2$ has two equal real roots.

(1) Find the expression of the function $f(x)$.
(2) For what values of $x$ is the graph of the function $f(x)$ above the graph of the function $g(x)$?
(3) Is there a pair of real numbers $m$, $n$ ($m < n$) such that the domain and range of the function $f(x)$ are respectively $[m, n]$ and $[2m, 2n]$? If so, find the values of $m$ and $n$, and explain the reasoning.

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