Binomials and Powers

Let $S (n) = \sum_{k = 0}^{n} (\binom{n}{k}) k^{n}$ .

You are given, $S (10) = 142469423360$ .

Find $S (10^{18})$ . Submit your answer modulo $83^{3} 89^{3} 97^{3}$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.