When working on complicated quantum information problems, the bra-ket notation can sometimes fail you, despite it being a handy tool. If your state vector includes a lot of different coefficients, writing out sequences can quickly become a very tedious task.
Luckily we can use a shortcut if we come across a maximally entangled state. In this video, Postdoctoral Researcher Ben Criger (QuTech) shows that every maximally entangled state is identical to one of the famous Bell states! That will simplify your derivation quite a bit. Besides its notational convenience, it also has an application for quantum information protocols, with which Dr. Criger will conclude the video.
How do you write the maximally entangled Bell state below in the X basis? and in the Y basis?
Read about the Bell states and how to create them in this article by Peter Young: https://young.physics.ucsc.edu/150/bell.pdf