
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 box contains $6$ cards, numbered with the integers $1$ to $6$. A card is randomly drawn from the box, its number $x$ is recorded, and then it is returned to the box. A second card is then drawn, and its number $y$ is recorded. Find the probability that:

$(\mathrm{I}) \ x + y$ is a multiple of $5$;

$(\mathrm{II}) \ x - y$ is a multiple of $3$;

$(\mathrm{III})$ At least one of $x$ or $y$ is $5$ or $6$.

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