This can't be a coincidence as well

Consider the decimal expansion of \(1 / 998998999\):

\[0. \quad 000 \quad 000 \quad 001 \quad 001 \quad 002 \\ \ \ \quad 004 \quad 007 \quad 013 \quad 024 \quad \ldots\]

Notice that those numbers in units of three digits - let's call them \(\lbrace a_{n} \rbrace\) - satisfy \(a_{n} = a_{n-1} + a_{n-2} + a_{n-3}\).

If \(a_{1} = a_{2} = 1\) and \(a_{3} = 2\), find the least \(n\) for which the recurrence is no longer satisfied.

Inspired by this problem.

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