Two identical rectangular prisms each have a height of 50 centimeters (cm). The base of each prism is a square, and the surface area of each prism is $P$ cm${}^2$. If the prisms are glued together along a square base, the resulting prism has a surface area of $\frac{41}{21}P$ cm${}^2$. What is the side length, in cm, of each square base?