
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
1. Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$, it has an extreme value of $10$ at $x=1$. Find the value of $a+b=\_\_\_\_\_\_\_\_$.
2. Calculate $\frac{A_{8}^{5}+A_{8}^{4}}{A_{9}^{6}-A_{9}^{5}} =\_\_\_\_\_\_\_\_\_.$
3. A rectangle with a perimeter of $20$ cm is rotated around one of its edges to form a cylinder. Find the maximum volume of the cylinder in $cm^{3}$.
4. If for a function $f(x)$ defined on $R$, for any two distinct real numbers $x_{1}$, $x_{2}$, it holds that $${x}_{1}f\left({x}_{1}\right)+{x}_{2}f\left({x}_{2}\right) > {x}_{1}f\left({x}_{2}\right)+{x}_{2}f\left({x}_{1}\right) $$, then the function $f(x)$ is called an "$H$ function". Among the following functions:

$①y=-{x}^{3}+x+1 ;②y=3x-2\left(\sin x-\cos x\right) ;③y={e}^{x}+1 ;④f\left(x\right)=\begin{cases}\ln \left|x\right|\;x\neq 0 \\\\ 0\;\;\;\;x=0\end{cases}$

Those that are "$H$ functions" are $\_\_\_\_\_\_\_\_$.

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