
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 set $A=\{x | x^2-2x-3 < 0\}$, $B=\{x | a-1 < x < a+1\}$, where $a\in \mathbb{R}$.

$(1)$ If $B \subseteq A$, find the set $C$ formed by the real number $a$;

$(2)$ Let the function $f(x)=4\sin x$, if the real number $x_0$ satisfies $f(x_0) \in C$, find the set of values for the real number $x_0$.

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