
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
Throw a uniform cubic die (with the numbers 1, 2, 3, 4, 5, 6 on its faces) twice in succession, and let the number that appears the first time be $x$ and the number that appears the second time be $y$.  
$(1)$ Calculate the probability of the event "$x+y\leqslant 3$";  
$(2)$ Calculate the probability of the event "$|x-y|=2$".

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