Bitter Numbers in the Bad Hall of Notoriety

Number Theory Level pending

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?

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