
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
In a bag, there are 2 black balls labeled $1$ and $2$, and 3 white balls labeled $3$, $4$, and $5$. These 5 balls are identical except for their labels and colors.  
$(1)$ If two balls are randomly drawn from the bag with replacement, one at a time, what is the probability of drawing a black ball first and then a white ball?  
$(2)$ If two balls are randomly drawn from the bag without replacement, denoting the label of a black ball as $x$ and the label of a white ball as $y$, what is the probability that $y-x \gt 2$?

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