Bitter Numbers in the Bad Hall of Notoriety

The president of Sugar Land has created this iterative rule for generating sequences of whole numbers:

1) If a number is $25$ or less, double that number.

2) If a number is greater than $25$ , subtract $12$ from it.

Let $S$ be the first number in a sequence created by the two rules above. $S$ is considered a "sweet" number if the sequence that starts with $S$ contains the number $16$ . If not, then $S$ is considered a bitter number and banned from Sugar Land. Furthermore, if the bitter number is divisible by $5$ , then it is placed in the Bad Numbers Hall of Notoriety in Bitter Nation. How many numbers from $1$ to $1,000$ are in the Bad Numbers Hall of Notoriety in Bitter Nation?