A very curious sequence

Algebra Level pending

Let us have the next recursive sequence of interger numbers \(a(n)\)

\(a(1)=1 \\ \\ a(n)=\begin{cases}-a(n/2), & \mbox { if } n \mbox { is even } \\ 1+a(n-1), & \mbox { if } n \mbox { is odd } \end{cases} \)

Find the minimum integer \(n\) such that \(a(n)=1000\). How many digits does it have?

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