
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 universal set $U=\mathbb{R}$, the domain of the function $f(x)= \frac{1}{ \sqrt{ \sin x- \frac{1}{2}}}+\lg (2-x^{2})$ is set $A$, and set $B=\{x|x^{2}-a < 0\}$.

1. Find the complement of set $A$ in $U$, denoted as $∁_{U}A$;
2. If $A∪B=A$, find the range of the real number $a$.

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