
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
Find all natural numbers  $n$  for which the following  $5$  conditions hold: $(1)$   $n$  is not divisible by any perfect square bigger than  $1$ . $(2)$   $n$  has exactly one prime divisor of the form  $4k+3$ ,  $k\in \mathbb{N}_0$ . $(3)$  Denote by  $S(n)$  the sum of digits of  $n$  and  $d(n)$  as the number of positive divisors of  $n$ . Then we have that  $S(n)+2=d(n)$ . $(4)$   $n+3$  is a perfect square. $(5)$   $n$  does not have a prime divisor which has  $4$  or more digits.

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