# By 40 it's over 10^(100000)

Computer Science Level 4

A sequence of positive integers $$S$$ with $$S_1=S_2=1$$ is further defined by $$S_k=S_{k-1}\times S_{k-2}+1$$ for $$k > 2.$$

The second time in the sequence that the last three digits of $$S_{n}$$ will equal the last three digits of $$S_{n+1}$$ will occur at $$S_{n=A},$$ whose last three digits are $$B.$$ Find $$A+B.$$

$$\textbf{Details and Assumptions}$$
The first time is the trivial case at $$S_{n=1}.$$
For example, if the last three digits of both $$S_{193}$$ and $$S_{194}$$ were both $$329,$$ then $$A=193$$ and $$B=329$$ so the answer would be $$522.$$

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