
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 a function $f(x) = -x^2 + mx + 1$ defined on the interval $[-1, 1]$, if there exists a point $x_0$ in the interval $(a, b)$ such that $f(x_0) = \frac{f(b) - f(a)}{b - a}$, then the function is said to have an "average value function" on the interval $[a, b]$, and $x_0$ is one of its mean points. Find the range of values for the real number $m$.

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