2019 AMC 12A Problem 19

In \triangle ABC with integer side lengths, \cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}. What is the least possible perimeter for \triangle ABC?

\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.