
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
For any positive integer  $k$  denote by  $S(k)$  the number of solutions  $(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+$  of the system  $$ \begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil  \mid x-y\mid =k , \end{cases} $$  where  $d$  is the greatest common divisor of positive integers  $x$  and  $y.$  Determine  $S(k)$  as a function of  $k$ . (Here  $\lceil z\rceil$  denotes the smalles integer number which is bigger or equal than  $z.$ )

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