
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
Define: A function $y=f(x)$ is called an "average value function" on the given interval $[a, b]$ within its domain if there exists an $x_0$ ($a < x_0 < b$) such that $f(x_0) = \frac {f(b)-f(a)}{b-a}$. The point $x_0$ is known as a mean value point. For example, $y=|x|$ is an average value function on $[-2, 2]$, with 0 being its mean value point. If the function $f(x) = x^2 - mx - 1$ is an "average value function" on the interval $[-1, 1]$, then determine the range of the real number $m$.

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