Recursion in between

Algebra Level pending

\[\large{{ a }_{ n+1 }=\left\lfloor \frac { 4 }{ 3 } \left( \left\lfloor \sqrt { { a }_{ n }^{ 2 }+5{ a }_{ n } } \right\rfloor \right) \right\rfloor }\]

Let there be a sequence defined by above rule for \(n\ge 1\). If \(a_{16}=628\) and the first perfect square immediately before \(a_{16}\) occurs at \(a_{x}\) (\(x<14\)) and \(\large{\left\lfloor \frac { { a }_{ 20 } }{ { a }_{ 10 } } \right\rfloor =y}\). Find \(x+y\)

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