Natural numbers $k, l, p$, and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$, then $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of $x^2 - px + q = 0$. What is the sum of all possible values of $q$?

Let's think step by step and output the final answer within \boxed{}.
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