2023 AMC 10A Problem 6

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21. What is the value of the cube?

\textbf{(A)} 42 \qquad \textbf{(B)} 63 \qquad \textbf{(C)} 84 \qquad \textbf{(D)} 126 \qquad \textbf{(E)} 252

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