AMC 12 2022b Problem 8

What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?

$\textbf{(A) }\ \text{two intersecting parabolas} \textbf{(B) }\ \text{two nonintersecting parabolas} \textbf{(C) }\ \text{two intersecting circles}$

$\textbf{(D) }\ \text{a circle and a hyperbola} \textbf{(E) }\ \text{a circle and two parabolas}$