Letter Chatters!

In a town with a population of 500 a person sends 2 letters to 2 random distinct people in the town, each of whom repeats the procedure. Thus, for each letter received, two letters are sent to 2 random distinct people. Note that if person A sends out a letter, he may send it out to any of the other 499 people, even those who are sending out letters at that time.

The probability that in the first \(12\) stages, the person who started the chain letter will not receive a letter is \( \alpha \). Find \([10^{10}\alpha]\).

Details and Assumptions

  • All letters reach their destination.
  • \([*]\) denotes the greatest integer function.

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