Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got $+50$ (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $[90, \infty)$ gets an A, a score in the range $[80, 90)$ gets a B, and a score in the range $[70, 80)$ gets a C. After the curve, Mr. Vader makes this statement: "Guess A, get an A. Guess B, get a B. Guess C, get a C." That is, answering every question with the answer choice $X$ would give, with the curve, a score receiving a grade of $X$, where $X$ is one of A, B, C. Luke, a student in Mr. Vader's class, was told ahead of time that there were either $5$ or $6$ answers as A on the test. Find the sum of all possible values of the number of questions on the test, given this information.

Let's think step by step and output the final answer within \boxed{}.
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