Consider the following equations for the Cipher feedback $($CFB$)$ mode:

Where I$0$ is the initialization vector with value IV$,$ Ii is the ith state of the shift register, and Pi is the plaintext, Ci is the ciphertext, Ek$($A$)$ is the encryption of plaintext A with key K$,$ and MSBs a function that returns the s highest bits of the block.

Formally prove that the mode works.$$ Please show and explain your logic of each step.

Also, please explain the advantages of OFB compared to CFB$.$