
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 $x \in (0,4)$, $y \in (0,4)$.

$(1)$ If $x \in \mathbb{N}^{*}$, $y \in \mathbb{N}^{*}$, and $x$, $y$ are used as the lengths of the sides of a rectangle, with the area of the rectangle denoted as $S$, find the probability that $S < 4$;

$(2)$ If $x \in \mathbb{R}$, $y \in \mathbb{R}$, find the probability that the difference between these two numbers is no greater than $2$.

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