Continuous Blackjack Threshold

A gambler draws Uniform(0,1) numbers and adds them up. He may stop at any point; if his total ever exceeds 1 he busts and loses. When he stops at total $s$, the dealer draws Uniform(0,1) numbers and adds them until the dealer's own total first exceeds $s$: the dealer wins if that total is at most 1, and the gambler wins otherwise.

Renewal theory gives the gambler's win probability when he stops at $s$: $W(s) = 1 - e^{s} + s,e^{s}.$

Optimal play is a threshold rule: keep drawing while the total is at most $t$, stop as soon as it exceeds $t$. Find $t$.

Answer format: round to 4 decimal places.