Let be the set of nonnegative integers. Define a function satisfying the following for any :
A happinization of a positive integer is an infinite sequence of integers, where the first term of the sequence is , and every term afterwards is the sum of the squares of the digits of the previous term. For example, the happinization of is . A happy number is a positive integer whose happinization ends with an infinite number of s; all other positive integers are sad numbers. For example, above is sad, but (with happinization ) is happy.
For any positive integer , define as the product of the first happy numbers. Let . Compute the value of .