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11 problems match “inclusion-exclusion”. Clear

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11. Possible second terms of a sequence ●●●●○ 100 ptsA sequence $a_1, a_2, \ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible?

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