
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 the sets $A=\{-1,0,1\}$ and $B=\{a+1,2a\}$. If $A \cap B = \{0\}$, then the value of the real number $a$ is \_\_\_\_\_\_\_\_.

(2) The domain of the function $f(x)=\sqrt{1-2^{x}}+\frac{1}{\sqrt{x+3}}$ is \_\_\_\_\_\_\_\_.

(3) Given the function $y=\sqrt{mx^{2}-6mx+m+8}$ has a domain of $\mathbb{R}$, then the range of the real number $m$ is \_\_\_\_\_\_\_\_.

(4) Given the function $f(x)=\begin{cases}x, & -1\leqslant x\leqslant 1 \\ 1-x, & x > 1 \text{ or } x < -1\end{cases}$. If $f(x)\geqslant \frac{1}{4}$, then the range of $x$ is \_\_\_\_\_\_\_\_.

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