The jumps of the bunny

A one year old bunny is sitting on the number 0 in the number line. His father, Bugs Bunny, is waiting for him on the number 10.

The bunny has to reach his father. At each minute, he can:

• Move one step forward;
• Move two steps forward;
• Stay still;
• Move one step backward;
• Move two steps backward.

There are \(N\) possible ways the bunny can be on the number 10 after 10 minutes. Find the last three digits of \(N\).

Details and assumptions

• The bunny is allowed to hop beyond 10, and then come back.

• There are no restrictions on the bunny entering the negative numbers. The bunny can go as much backwards as he wants.

• Order does matter. For example, the steps \(\{0000022222\}\) and \(\{2020202020\}\) are considered distinct.