Consider a perfect cube $N^3$ where $N$ is a positive integer. You are given the following cryptic clues related to $N$:
- The sum of the digits of $N$ is a perfect square.
- $N$ is divisible by the sum of its digits.
- The prime factorization of $N$ includes exactly three distinct primes.
Determine the smallest possible value of $N$ that satisfies all these conditions.
$\textbf{My Work:}$
Let $N = abc$ be the three-digit number representing the cube. Without loss of generality, assume $a$, $b$, and $c$ are the digits of $N$.
Despite my efforts in analyzing the conditions and attempting various approaches, I couldn't determine the smallest possible value of $N$ that satisfies all these conditions. Any guidance, insights, or a solution from the community would be greatly appreciated.