Add Again?

Aahna plays the following game: She generates a uniformly random number $U_1 \sim \unif{0}{1}$. 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 $U_2 \sim \unif{0}{1}$, and she adds it to her balance. If her balance ever exceeds $1$, she loses and receives no payout. Otherwise, while her balance is less than $1$, she faces the same option to hit or hold. Assuming Aahna plays optimally, she will stop the first time her balance exceeds $t$ for some $0 < t < 1$. Find $t$ to the nearest thousandth.