2022 AMC 12A Problem 16

A triangular number is a positive integer that can be expressed in the form t_n=1+2+3+\cdots+n, for some positive integer n. The three smallest triangular numbers that are also perfect squares are t_1=1=1^2, t_8=36=6^2, and t_{49}=1225=35^2. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?

\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27

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