\[ \large \left \lfloor \frac{a^2}b \right \rfloor + \left\lfloor \frac{b^2}a \right \rfloor = \left \lfloor \frac{a^2+b^2}{ab} \right \rfloor + ab \]

Given \(a\) and \(b\) are positive integers with \(a<100\), if the ordered pairs \((a_1, b_1)\), \((a_2, b_2)\), \(\ldots\) , \((a_k, b_k)\) satisfy the equation above, evaluate \(\displaystyle \sum_{i=1}^{k} (a_i+b_i)\).

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