Let $A=\{1,2,3, \ldots, 100\}$ and $R$ be a relation on $A$ such that $R=\{(a, b): a=2 b+1\}$. Let $\left(a_1, a_2\right),\left(a_2, a_3\right),\left(a_3, a_4\right), \ldots .,\left(a_k, a_{k+1}\right)$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$, for which such a sequence exists, is equal to :