Suppose the Cookie Monster has two (distinguishable) jars of cookies, each of which initially contains $$20$$ cookies. The Monster then starts to eat the cookies one at a time such that each time he takes a cookie from a jar chosen uniformly at random.

Now the Cookie Monster never looks into the jars, so he never knows how many cookies are left in either jar until he reaches in to one and finds that that jar is empty. When he first discovers that one of the jars is in fact empty, the probability that there are (strictly) fewer than $$3$$ cookies in the other jar is $$S.$$

Find $$\lfloor 1000S \rfloor$$.

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