
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 $f(x)=x^{3}-2f{{'}}(1)x$, find $f{{'}}(1)$.
2. The function $f(x)=a^{2}x-2a+1$ is given. If the statement "$\forall x\in (0,1),f(x)\ne 0$" is false, find the range of values for $a$.
3. In $\triangle ABC$, $AC=\sqrt{2}$, $BC=\sqrt{6}$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$. If there exists a point $D$ on the extension of line segment $BA$ such that $\angle BDC=\frac{\pi }{4}$, find $CD$.
4. The function $f(x)=\begin{cases} & {{x}^{2}}+x+a,x < 0 \\\\ & -\frac{1}{x},x > 0 \\end{cases}$ is given. If there exist two distinct points $A$ and $B$ on the graph of $y=f(x)$ such that the tangent lines at these points overlap, find the range of values for the real number $a$.

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