2025 AIME II Problem 14

Let {\triangle ABC} be a right triangle with \angle A = 90^\circ and BC = 38. There exist points K and L inside the triangle suchAK = AL = BK = CL = KL = 14.The area of the quadrilateral BKLC can be expressed as n\sqrt3 for some positive integer n. Find n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

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