A very curious sequence

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?