Factorization is an art(2)

Algebra Level 5

\[\large \begin{cases} a = x+5x^{\frac{1}{5}}y^{\frac{4}{5}}+10x^{\frac{3}{5}}y^{\frac{2}{5}} \\ b = y+5x^{\frac{4}{5}}y^{\frac{1}{5}}+10x^{\frac{2}{5}}y^{\frac{3}{5}} \\ c = (a+b)^{\frac{2}{5}}+(a-b)^{\frac{2}{5}} \\ x^{\frac{2}{5}}+y^{\frac{2}{5}}=1008 \end{cases} \]

Given the above, find \(c\).

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