Each round consists of flipping all of the coins showing heads.

If an even number of coins is flipped and an even number of heads comes up the player loses.

If an odd number of coins is flipped and an odd number of heads comes up the player loses.

Rounds continue until the player either loses or reaches zero heads.

The player wins by getting to zero heads without losing.

What is the probability of winning this game?

Examples:
6→3→1 would be a loss.
6→3→0 would be a win.
6→0 would be a loss (even→even rule is applied first.)