Is it too late for April Fools?

Algebra Level 5

Let \(\mathbb{N}\) be the set of nonnegative integers. Define a function \(f : \mathbb{N}^3 \to \mathbb{N}\) satisfying the following for any \(a,b,c \in \mathbb{N} \setminus \{0\}\):

\[\begin{cases} f(0,0,0) &= 2015 \\ f(a,0,0) &= f(a-1,1,0) \\ f(0,b,0) &= f(0,b-1,1) \\ f(0,0,c) &= f(c-1,0,0) \\ f(a,b,0) &= f(a-1,b-1,2) \\ f(a,0,c) &= f(c-1,1,a-1) \\ f(0,b,c) &= f(0,b-1,c+1) \\ f(a,b,c) &= f(b,c,a-1) \\ \end{cases}\]

A happinization of a positive integer \(n\) is an infinite sequence of integers, where the first term of the sequence is \(n\), and every term afterwards is the sum of the squares of the digits of the previous term. For example, the happinization of \(2015\) is \(2015, 30, 9, 81, 65, 61, 37, 58, 89, 145, 20, 4, 16, 37, \ldots\). A happy number is a positive integer whose happinization ends with an infinite number of \(1\)s; all other positive integers are sad numbers. For example, \(2015\) above is sad, but \(1\) (with happinization \(1, 1, 1, 1, \ldots\)) is happy.

For any positive integer \(n\), define \(p_n\) as the product of the first \(n!\) happy numbers. Let \(A = p_1, F = p_2, I = p_3, L = p_4, O = p_5, P = p_6, R = p_7, S = p_8\). Compute the value of \(f(APRIL,FOOLS,2015)\).

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