
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 $|\vec{a}|=4$, $|\vec{b}|=1$, and the angle between $\vec{a}$ and $\vec{b}$ is $\theta$, and $|\vec{a}-2\vec{b}|=4$, then the value of $\cos\theta$ is ______.

(2) The general term of the sequence $\{a_n\}$ is $a_n=\frac{1}{\sqrt{n}+\sqrt{n+1}}$, then the sum of the first 8 terms of this sequence equals ______.

(3) The solution set of the inequality $\frac{x^2-1}{2x-1}\leqslant 0$ is ______.

(4) Given vectors $\vec{a}=(m,1)$, $\vec{b}=(4-n,2)$, $m>0$, $n>0$, and $\vec{a}\parallel\vec{b}$, then the minimum value of $\frac{1}{m}+\frac{8}{n}$ is ______.

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