
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) The solution set of the inequality $|8-3x| > 0$ is ______________.

(2) The solution set of the inequality $x^{2}-3|x|+2\leqslant 0$ is ______________.

(3) If positive real numbers $a$ and $b$ satisfy $ab=2$, then the minimum value of $(1+2a)\cdot(1+b)$ is ______________.

(4) Given vectors $a=(x-1,2)$ and $b=(4,y)$, if $a\perp b$, then the minimum value of $9^{x}+3^{y}$ is ______________.

(5) The graph of the function $y=a^{1-x} (a > 0,a\neq 1)$ always passes through a fixed point $A$. If point $A$ lies on the line $mx+ny-1=0 (mn > 0)$, then the minimum value of $\dfrac{1}{m}+ \dfrac{1}{n}$ is ______________.

(6) For any real number $x\in R$, the inequality $x^{2}+a|x|+1\geqslant 0$ always holds, then the range of the real number $a$ is ______________.

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