Greedy Squares

Consider the following recursive process:

\[a_{n+1}=a_n-\lfloor\sqrt{a_n}\rfloor^2\]

For all positive integers \(a_0\), the process above eventually results in \(a_n = 0\) for all \(n \geq k\) for some positive integer \(k\). What is the first value of \(a_0\) such that \(k=7\text{?}\)

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