
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
Definition: If the function $y=f(x)$ exists in the given interval $\left[a,b\right]$ within its domain, and there exists $x_{0}$ $(a < x_{0} < b)$ such that $f(x_{0})=\frac{f(b)-f(a)}{b-a}$, then the function $y=f(x)$ is called the "average value function" on $\left[a,b\right]$, and $x_{0}$ is its average value point.  
$(1)$ Is the function $y=2x^{2}$ an "average value function" on $\left[-1,1\right]$? If yes, find its average value point. If not, explain the reason.  
$(2)$ Given the function $y=-2^{2x+1}+m\cdot 2^{x+1}+1$ is an "average value function" on $\left[-1,1\right]$, find the range of real number $m$.

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