A good sequence

A sequence of real numbers \(a_0, a_1, a_2 \ldots\) is said to be good if the following three conditions hold.

  • The value of \(a_0\) is a positive integer.
  • For each non-negative integer \(i\) we have \(a_{i+1}=2a_i+1\) or \(a_{i+1}=\dfrac{a_i}{a_i+2}\).
  • There exists a positive integer \(k\) such that \(a_k=2014\).

Find the smallest positive integer \(n\) such that there exists a good sequence \(a_0, a_1, a_2 \ldots \) of real numbers with the property that \(a_n=2014\).


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