Passing Balls: Hard Version

Computer Science Level pending

The binary tree above has a height of 2 and has 4 leaves. You are going to pass down a ball from the root (the topmost node). Every node other than the leaves has a switch. If a node is switched off, the ball will pass to its left child; if a node is switched on, the ball will pass to its right child. After passing the ball, the switch will toggle (if it's on it becomes off and vice versa). Initially, every node is switched off.

Chris built a similar tree of height 100 with the same indexing. Which leaf will contain the \(3366^\text{th}\) ball?

Output you answer modulo 1000.

You are advised to solve the Easy and Medium version of this problem first.

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