
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 function $f(x)$, if there exists a real number $x$ within its domain such that $f(-x) = -f(x)$, then $f(x)$ is called a "locally odd function."

(I) Given the quadratic function $f(x) = ax^2 + 2bx - 3a$ ($a, b \in \mathbb{R}$), determine whether $f(x)$ is a "locally odd function" and explain the reasoning;

(II) Suppose $f(x) = 2^x + m - 1$ is defined on the interval $[-1, 2]$ and is a "locally odd function", find the range of the real number $m$;

(III) Suppose $f(x) = 4^x - m \cdot 2^{x+1} + m^2 - 3$, if $f(x)$ is not a "locally odd function" defined on the domain $\mathbb{R}$, find the range of the real number $m$.

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