\(+\) and \(\times\) defined
Define \(S(n)\) to be the term directly succeeding \(n\) in A000027. For example, \(S(5) = 6\) because the term succeeding \(5\) in the sequence is \(6\). Also define \(S(0) = 1\). \(S\) is undefined everywhere else, and you may assume that \(S\) is injective.
Define the operation \(+\) as follows:
- \(a + 0 = a\)
- \(a + S(b) = S(a+b)\)
Define the operation \(\times\) as follows:
- \(a \times 0 = 0\)
- \(a \times S(b) = a + (a \times b)\)
Determine the value of \(116180912 \times 615151219\).