Recovering starting stacks in a seven-player doubling game

Seven players take turns in order, each flipping a coin once. On heads, the flipper must pay each of the other six players an amount equal to that player's current stack, doubling it. All seven flips come up heads, and afterwards each player holds exactly $1.28. If $x_1,\dots,x_7$ are the players' starting stacks in dollars in flipping order, find $10000\left(x_1^2+\cdots+x_7^2\right)$.

$269374$. Work backwards from $1.28 each: to undo a payout, halve everyone except the payer and return the halves to the payer. The total $8.96 is conserved, and the starting stacks come out as $4.49, 2.25, 1.13, 0.57, 0.29, 0.15, 0.08$, whose squares times $10000$ sum to $269374$.