Chocolate

Anna and Beka are dividing a rectangular chocolate bar of size M \times N. They take turns breaking off a square piece from one end of the chocolate bar, with the side length equal to the shorter side of the bar. If at any point the remaining chocolate is square-shaped, the child whose turn it is takes the entire remaining piece, and the division ends. Anna goes first.

Consider an example: the chocolate bar is initially 6 \times 10. Anna breaks off a 6 \times 6 piece. Beka then takes a 4 \times 4 piece from the remaining 6 \times 4 chocolate. Anna next takes a 2 \times 2 piece from the remaining 2 \times 4, and Beka takes the final 2 \times 2 piece. Thus, Anna receives pieces with a total area of $40$, and Beka receives pieces with a total area of $20$.

Given the total area of chocolate each child received, determine the initial dimensions $M$ and $N$ of the chocolate bar such that the areas received by the children match the given values.

If there are multiple solutions, choose the one where $M$ is smallest. If there are still multiple solutions, choose the one where $N$ is smallest. If no such $M$ and $N$ exist, output two numbers $-1$.