How would you solve the diophantine $x^4+y^4=2z^2$

I would like to know any way of solving the diophantine equation $x^4+y^4=2z^2$. Or ideas that seem worth trying out.

By solving I mean fining all solutions and proving there are no more.

Keith Conrad showed how to reduce this equation to a different one which was solved by Fermat in his notes about Fermat descent, other than I have no ideas how to solve it. I tried to do descent on it directly but that seems completely impossible so I am interested in other techniques. Thanks very much.

Solutions Collecting From Web of "How would you solve the diophantine $x^4+y^4=2z^2$"