You have N bags. Bag 1 has a black ball, Bag 2 has a black ball and a white ball, Bag 3 has a black ball and two white balls, and so on. Bag N has a black ball and N-1 white balls. You pick a ball from each bag at random and record the numbers of the bags that you picked a black ball from. For example, if you had 100 bags, then your sequence might be 1, 2, 3, 10, 14, 37. Call the last number in your sequence X. Prove that X is a random number from 1 to N with a uniform distribution.