A game is played starting with 6 fair coins laid out with heads face up.

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

If fewer than half of the flipped coins come up heads the player loses.

Rounds continue until the player either loses or has one heads remaining.

The player wins by getting to one heads without losing.

What is the probability of winning this game?

Examples:
6→4→1 would be a loss. (1 is less than half of 4.)
6→5→3→2→2→2→1 would be a win.

If I didn't make a mistake, then when you have 4 heads and you flip and 3 heads come up, your chances of winning actually go down, going from 12/35 (34.29%) to 2/7 (28.57%), (i.e doesn't improve). Probably because you are now starting with an odd number of heads, so you have a 50% chance of losing immediately on the very next turn. Good "strategy" is to avoid an odd number of heads until the very end of the game.