
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 $x$ and $y$ be positive integers such that $1 \leqslant y<x$. The pair $(x, y)$ is called "nice" if $x$ and $y$ have exactly $x-y$ positive common divisors. For example, 30 and 42 have exactly 4 common divisors: $1, 2, 3,$ and $6$. Since $42-30=12 \neq 4$, the pair $(42,30)$ is not nice. However, the pair $(8,6)$ is nice, because 6 and 8 have $8-6=2$ positive common divisors: $1$ and $2$.

For any integer $n \geq 2$, let $s(n)$ denote the number of pairs of integers $(x, y)$ such that $1 \leq y < x \leq n$ and $(x, y)$ is nice.

a) Is there an integer $n \geq 2$ such that $s(n)=2022$?

b) Is there an integer $n \geq 2$ such that $s(n)=2023$?

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