2022 Cayley Problem 23

Andreas, Boyu, Callista, and Diane each randomly choose an integer from 1 to 9, inclusive. Each of their choices is independent of the other integers chosen and the same integer can be chosen by more than one person. The probability that the sum of their four integers is even is equal to \dfrac{N}{6561} for some positive integer N. What is the sum of the squares of the digits of N?

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Full credit to this problem is given to the CEMC, you may view all Cayley contests here.