Problem 19 (2022 AMC 12B)

In triangle ABC, the medians AD and BE intersect at G, and triangle AGE is equilateral. The lengths AB = 13, BC = 14, and CA = 15. Find the value of m + n + p if the side length of triangle AGE can be written as (m/n)√p, where m and n are relatively prime positive integers and p is a positive integer not divisible by the square of any prime.