To play the game of “Zactoll” (named for yours truly, of course), you start with the number 1 and then try to reach a target number through a series of steps. For each step, you can always choose to double the number you currently have. However, if the number happens to be one more than an odd multiple of 3, you can choose to “reduce” — that is, subtract 1 and then divide by 3.
For example, to reach a target number of 5, you could double (1 → 2), double (2 → 4), double (4 → 8), double (8 → 16) and reduce (16 → 5). And to get a target number of 6, you could double (1 → 2), double (2 → 4), double (4 → 8), double (8 → 16), reduce (16 → 5), double (5 → 10), reduce (10 → 3) and double (3 → 6).
What is the smallest target number you cannot reach in the game of Zactoll?
"Zactoll" is an anagram of Collatz, and seems to be somewhat of an inverse implementation of the Collatz Conjecture.
If there is a smallest unreachable target number, it must be greater than $19 \cdot 2^{58}$.