AMC 12 2019a Problem 19

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

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