2022 AMC 12B Problem 19

In \triangle{ABC} medians \overline{AD} and \overline{BE} intersect at G and \triangle{AGE} is equilateral. Then \cos(C) can be written as \frac{m\sqrt p}n, where m and n are relatively prime positive integers and p is a positive integer not divisible by the square of any prime. What is m+n+p?

\textbf{(A) }44 \qquad \textbf{(B) }48 \qquad \textbf{(C) }52 \qquad \textbf{(D) }56 \qquad \textbf{(E) }60

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