
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 $f(x)=x^{3}+2xf'(1)$, then $f'(1)=$ ______.

(2) Given $p: |x-a| < 4$, $q: x^{2}+5x-6 > 0$, and $q$ is a sufficient but not necessary condition for $p$, then the range of values for $a$ is ______.

(3) Given the function $f(x)= \frac{2}{x}+\ln x$, consider the following four propositions:  
(i) $x=2$ is a local minimum point of $f(x)$;  
(ii) The function $f(x)$ has a unique zero in $(0,+\infty)$;  
(iii) There exists a positive real number $k$, such that $f(x) > kx$ always holds;  
(iv) For any two positive real numbers $x_{1}$, $x_{2}$, and $x_{1} < x_{2}$, if $f(x_{1})= f(x_{2})$, then $x_{1}+x_{2} > 4$.  
Among these propositions, the true ones are ______.

(4) Given $f(x)$ is an even function defined on $\mathbb{R}$ and satisfies $f(x+2)=- \frac{1}{f(x)}$ for $2\leqslant x\leqslant 3$, when $f(x)=x$, then $f(- \frac{11}{2} )=$ ______.

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