Statement

William has recently built a Water Calculator^, which mimics the operation of a computer through water pipes, sinks and siphons. Due to the rudimentary nature of its circuits, the Water Calculator^ is able to store a single number up to a billion and perform only a handful of atomic operations: (1) add or subtract 1; (2) multiply by 2; (3) divide by a power of 3 (which divides exactly the stored number). These few operations are anyway more than sufficient for William's basic mathematical needs. In this moment, the Water Calculator^ is holding number N. Since William needs to leave his home for a while, it is crucial to empty the calculator in order to avoid malicious rusting! Calculate the minimum number of atomic operations necessary for William to completely empty the calculator.