Numbers and numbers

Algebra Level 5

Let \(S\) be the set of all ordered triple of integers \((a_1,a_2,a_3)\) with \(1 \le a_1,a_2,a_3 \le 10\). Each ordered triple in \(S\) generates a sequence according to the rule \(a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |\) for all \(n\ge 4\). Find the number of such sequences for which \(a_n=0\) for some \(n\).

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