
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
A fair die is rolled twice in succession, and the numbers facing up are observed and recorded as $x$ and $y$ respectively.  
$(1)$ If the event "$x+y=8$" is denoted as event $A$, find the probability of event $A$ occurring;  
$(2)$ If the event "$x^{2}+y^{2} \leqslant 12$" is denoted as event $B$, find the probability of event $B$ occurring.

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