
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) $\int_{0}^{1}(\sqrt{1-(x-1)^{2}}-2x)dx=$ ______.

(2) Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with common difference $d$, then the sequence $S_{6}-S_{3}, S_{9}-S_{6}, S_{12}-S_{9}$ is an arithmetic sequence with a common difference of $9d$. By analogy, we can conclude: Let $T_{n}$ be the product of the first $n$ terms of a geometric sequence $\{b_{n}\}$ with a common ratio of $2$, then the sequence $\frac{T_{6}}{T_{3}}, \frac{T_{9}}{T_{6}}, \frac{T_{12}}{T_{9}}$ is a geometric sequence, and its common ratio is ______.

(3) The function $f(x)=ax^{3}-3x$ is monotonically decreasing on the interval $(-1,1)$, then the range of $a$ is ______.

(4) Let the function $f(x)$ have a derivative $f'(x)$ in its domain $D$. If there exists a real number $a$ and a function $h(x)$, where $h(x) > 0$ for any $x \in D$, such that $f'(x)=h(x)(x^{2}-ax+1)$, then the function $f(x)$ is said to have property $P(a)$. Given the following four functions:  
① $f(x)=\frac{1}{3}x^{3}-x^{2}+x+1$;  
② $f(x)=\ln x+\frac{4}{x+1}$;  
③ $f(x)=(x^{2}-4x+5)e^{x}$;  
④ $f(x)=\frac{x^{2}+x}{2x+1}$,  
among them, the functions that have property $P(2)$ are ______. (Write down all the numbers of the functions that meet the condition)

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