
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) If $f(x)=x^{2}-2x-4\ln x$, then the decreasing interval of $f(x)$ is $\boxed{\left(0,2\right)}$.

(2) Let $z_{1},z_{2}$ be complex numbers, and consider the following four statements:

(i) If $|z_{1}-z_{2}|=0$, then $\overline{z_{1}}=\overline{z_{2}}$;  
(ii) If $z_{1}=\overline{z_{2}}$, then $\overline{z_{1}}=z_{2}$;  

(iii) If $|z_{1}|=|z_{2}|$, then $z_{1}\cdot \overline{z_{1}}=z_{2}\cdot \overline{z_{2}}$;  
(iv) If $|z_{1}|=|z_{2}|$, then $z_{1}^{2}=z_{2}^{2}$.

Among these, the true statements are $\boxed{\text{(i)(ii)(iii)}}$.

(3) Given that the function $f(x)=x(x-c)^{2}$ has a local minimum at $x=2$, the value of the real number $c$ is $\boxed{2}$.

(4) If the curves $y=x^{2}-1$ and $y=a\ln x-1$ have a common tangent line, then the range of values for the positive real number $a$ is $\boxed{(0,2e]}$.

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