Five hundred infinitesimal ants are distributed independently and uniformly at random along a 1-foot string. Each ant immediately chooses a direction (left or right, with equal probability and independently) and moves along the string at 1 foot per minute. If two ants collide head-on, they instantly reverse directions and keep going. When an ant reaches either end, it falls off. What is the expected time for all ants to fall off the string?
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