Algebra Level 5

Let $$\mathbb{S}$$ be the set $$\mathbb{R}^+ \cup \{0\}$$

A function $$f:\mathbb{S} \rightarrow \mathbb{S}$$ is defined as -$f(x^2+y^2) = y^2f(x)+x^2f(y) +x^4+y^4$

Then the value of $$f(2015)$$ can be written as $$a^d b^d c^d$$ where $$a,b,c,d$$ are all distinct prime numbers. Find the value of $$a+b+c+d$$.

Give it a thought -

$$\bullet$$ Will the answer change if $$\mathbb{S}$$ was replaced by $$\mathbb{R}$$ (Real numbers) or $$\mathbb{C}$$ (Complex numbers)?

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