
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 $f(x)=x^{3}-2f'(1)x$, then $f'(1)=$_______

(2) Given the function $f(x)=a^{2}x-2a+1$, "If $\forall x \in (0,1)$, $f(x) \neq 0$" is a false statement, then the range of $a$ is_______

(3) Given in $\Delta ABC$, $AC=\sqrt{2}$, $BC=\sqrt{6}$, the area of $\Delta ABC$ is $\frac{\sqrt{3}}{2}$. If there exists a point $D$ on the extension line of $BA$ such that $\angle BDC=\frac{\pi }{4}$, then $CD=$_______

(4) For a continuous function $f(x)$ defined on $\mathbb{R}$, which is an odd function, its derivative is $f'(x)$, and for any $x$ in its domain, $2f(x) < x[3-f'(x)]$ holds, then the solution set of the inequality $x^{2}f(x)-9f(3) < x^{3}-27$ is_______

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