I'm exploring a mathematical process that involves three operations (D, V, W) applied to odd integers, starting from $1$. Here's how these operations are defined:
- D: $ i' = 4i + 1 $
- V: $ i' = \frac{2i - 1}{3} $ (only valid if $i' $ is an odd integer)
- W: $ i' = \frac{4i - 1}{3} $ (only valid if $ i' $ is an odd integer)
The operation D is always valid. Sometimes, either D or V, or D or W may be applicable. One of these operations is randomly selected among the valid ones for each iteration. There are no limits on the number of iterations or the number of attempts, starting anew from $1$ each time.
For example, the odd number $1218886684813$ can be reached from $1$ using the sequence of choices DDDDVDDWWDWVDDDWDDWDDDDDVDDVD.
My question is: Can every odd integer be reached using this procedure, given no constraints on the number of iterations or trials?