
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 that the domain of the function $f(x)$ is $(0, +\infty)$, if $y = \frac {f(x)}{x}$ is an increasing function on $(0, +\infty)$, then $f(x)$ is called a "first-order ratio-increasing function"; if $y = \frac {f(x)}{x^2}$ is an increasing function on $(0, +\infty)$, then $f(x)$ is called a "second-order ratio-increasing function." We denote the set of all "first-order ratio-increasing functions" as $\Omega_1$, and the set of all "second-order ratio-increasing functions" as $\Omega_2$. Given the function $f(x) = x^3 - 2mx^2 - mx$, if $f(x) \in \Omega_1$ and $f(x) \notin \Omega_2$, find the range of the real number $m$.

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