Bouncing particles leaving a one-metre rod

A segment one metre long carries $1000$ point particles, each placed at an independent uniformly random position and each moving left or right at $1$ metre per second. Whenever two particles meet they instantly reverse direction, keeping the same speed. A particle leaves when it reaches an end of the segment. What is the expected time until every particle has left?

$1000/1001$ seconds. Swapping labels at each collision shows the particles behave as if they pass through one another. Each 'ghost' particle exits after a time equal to its distance to the end it faces, which is uniform on $[0,1]$. The last exit is the maximum of $1000$ independent uniforms, with mean $\tfrac{1000}{1001}$.