Robot Takeoff Threshold

A robot runs down a track from 0 towards a takeoff line at 1. Each advance draws a real number uniformly from $[0,1]$ and adds it to the robot's position.

• If an advance takes the robot to 1 or beyond, the attempt scores 0.
• Otherwise the robot may jump: it draws one final uniform number from $[0,1]$, adds it to its current position, and that sum is the score. Or it may advance again.

The robot uses a threshold rule: jump as soon as its position is at least $t$. Simulating 400,000 attempts per threshold gives

threshold $t$	0.15	0.30	0.45	0.60
mean score	1.062	1.087	1.058	0.949

Compute the exact threshold that maximises the expected score.

Answer format: round to 3 decimal places.