Dorothy has a mysterious button on her calculator. When she types in an integer and hits the button,
if the input is odd, the calculator outputs less than triple the input.
if the input is even but not divisible by , the calculator outputs more than half the number.
if the input is divisible by , the calculator outputs one-fourth of the input.
Dorothy typed in an integer, hit the button, and saw an output of . What are all possible integers Dorothy may have input?