Aahna plays the following game: She generates a uniformly random number . She has two options: Hit or Hold. If she decides to hold, she leaves with her current balance. Otherwise, if she hits, she generated another uniformly random number , and she adds it to her balance. If her balance ever exceeds , she loses and receives no payout. Otherwise, while her balance is less than , she faces the same option to hit or hold. Assuming Aahna plays optimally, she will stop the first time her balance exceeds for some . Find to the nearest thousandth.