Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Express $f(n+1)-f(n-1)$ in terms of $f(n)$. The original answer is in the form of a single variable k, please provide the value of k.

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