
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
Given the function $f(x)=\sqrt{3-x}+\frac{1}{{\sqrt{x}}}$ with the domain $A$ and the set $B=\{x\left|\right.1-a \lt x\leqslant 2a+4\}\left(a \gt -1\right)$.<br/>$(1)$ If $a=0$, find $A\cap B$ and $A\cup B$;<br/>$(2)$ If the proposition "$\forall x\in A$, $x\in B$" is true, find the range of the real number $a$.

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