
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
Let the function $y=\sqrt{x+1}$ have a domain set $A$, and the solution set of the inequality $\log_{2}(x-1) \leqslant 1$ be set $B$.

$(1)$ Find the sets $A,B$;        $(2)$ Find the sets $A \cup B, A \cap C_{R}B$.

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