Barely an Integer!

\[\large{n = \left \lfloor m \right \rfloor + \left \lfloor \dfrac{m}{2!} \right \rfloor + \left \lfloor \dfrac{m}{3!} \right \rfloor + \ldots + \left \lfloor \dfrac{m}{k!} \right \rfloor + \ldots }\]

For every natural number \(n\), define \(S(n)\) to be the unique integer \(m\) (if it exists) which satisfies the equation above. Submit the value of \(S(3438)\) as your answer.


Bonus: Does there exist a number \(k\) such that, for any non-negative integer \(n\), at least one of \(S(n+1), \ S(n+2),\ \ldots \ , \ S(n+k)\) exists?

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