2022 AIME I Problem 14

Given \triangle A B C and a point P on one of its sides, call line \ell the splitting line of \triangle A B C through P if \ell passes through P and divides \triangle A B C into two polygons of equal perimeter. Let \triangle A B C be a triangle where B C=219 and A B and A C are positive integers. Let M and N be the midpoints of \overline{A B} and \overline{A C}, respectively, and suppose that the splitting lines of \triangle A B C through M and N intersect at 30^{\circ}. Find the perimeter of \triangle A B C.