Reciprocal fraction

Algebra Level 3

Consider the sequence $a_{1}=3$, $a_{2}$, $a_{3} , \ldots$, where $\dfrac{1}{a_{k+1}}=\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{k}} \text{ for all integers } k \ge 1.$

Find $a_{2016}$.

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