Recurring Styles #7 - Perfect Squares!

\[\large{y_{n+2} = (4k-5)y_{n+1} - y_n + 4 - 2k \ , \quad n \geq 1}\]

Define the sequence \((y_n)_{n=1}^\infty\) above by \(y_1 = y_2 = 1\). Find the sum of all integers \(k\) such that every term of the sequence is a perfect square.

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