For positive integers \(n,\) the double factorial function is defined as \[ n!! = \begin{cases} n(n-2)(n-4)\ldots\cdot4\cdot2, \; n \; \text{even} \\ n(n-2)(n-4)\ldots\cdot3\cdot1, \; n \; \text{odd} \end{cases}\] The ratio \(\dfrac{513!}{513!!}\) can be written as \(2^a \cdot b!\) for positive integers \(a\) and \( b\) in several different ways. For the smallest value of \(b\), what is the corresponding value of \(a\)?

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