Come and See this Nice Real System

Algebra Level 5

Let \((a_1,b_1)\), \((a_2,b_2)\), ..., \((a_n,b_n)\) be all the pairs of real numbers that satisfy following system of equations:\[ \left\{\begin{matrix} a^3+6ab+8= b^3 \\a^5+32= b^5 \end{matrix}\right. \] Find the value of\[n+\sum_{k=1}^{n} (a_k^2+b_k^2) \ .\]

This problem is original and belongs to this set.
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