# $$+$$ and $$\times$$ defined

Algebra Level 4

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 $$116180912 \times 615151219$$.

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