
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
Let the function $y=f(x)$ have a domain $D$, and given an interval $\left[a,b\right]\subseteq D$. If there exists $x_{0}\in \left(a,b\right)$ such that $f(x_{0})=\frac{f(b)-f(a)}{b-a}$, then the function $y=f(x)$ is called the "mean function" on the interval $\left[a,b\right]$, and $x_{0}$ is called the "mean point" of the function.  
$(1)$ Determine whether the function $y=x^{2}$ is a "mean function" on the interval $\left[1,2\right]$. If it is, find its "mean point"; if not, explain the reason.  
$(2)$ Given that the function $y=-2^{2x-1}+m\cdot 2^{x-1}-12$ is a "mean function" on the interval $\left[1,3\right]$, find the range of real numbers for $m$.  
$(3)$ If the function $y=\frac{{x}^{2}+a}{2({x}^{2}-2x+2)}$ (where $a$ is a constant in $\mathbb{R}$) is a "mean function" on the interval $\left[-2,2\right]$, and $\frac{2}{3}$ is its "mean point". Divide the interval $\left[-2,0\right]$ into $m+1$ parts, where $m\in \mathbb{N}$, with partition points $t_{1}$, $t_{2}$, ..., $t_{m}$, and $t_{0}=-2$, $t_{m+1}=0$. Define $G=\sum_{i=0}^{m}{|f({t}_{i+1})-f({t}_{i})|}$. Divide the interval $\left[0,2\right]$ into $2^{n}+1$ parts, where $n\in \mathbb{N}$, with partition points ${x}_{1}$, ${x}_{2}$, ..., ${x}_{{2}^{n}}$, and define $H=\sum_{i=1}^{{2}^{n}}{f({x}_{i})}$. Find the smallest integer value of $n$ such that $H\cdot G \gt 2023$.

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