
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 $f(x) = 2x^2 + mx - 1$, where $m$ is a real number.  
(1) It is known that for any real number $f(x)$, $f(x) = f(2-x)$ holds. Let the set $A = \{y | y = f(x), x \in \left[-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right]\}$, find the set $A$.  
(2) Denote the set of all negative numbers as $R^{-}$, and $R^{-} \cap \{y | y = f(x) + 2\} = \emptyset$, find the set of all $m$ that satisfy the condition;  
(3) Let $g(x) = |x-a| - x^2 - mx$ ($a \in R$), find the minimum value of $f(x) + g(x)$.

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