Keep or Play: Optimal Replay in a Coin-Flipping Game

You are offered the following game. You flip a fair coin repeatedly until you land tails. For each head you flip before that first tails, you earn $\$h$. Once tails appears, you can either keep your payout or abandon it and play a second round under the same rules. If you replay, your first-round earnings are forfeited and you must accept whatever the second round gives you (no further replays). Find the optimal strategy and the fair value of this game when $h = 10$.