Does it help to know that 6 - 5 = 1 ?

Algebra Level 3

It is given that $x,y\in\Bbb{Z^+}$ and they satisfy the equation given below:

$\large\prod_{k=0}^5\left(5^{2^k}+6^{2^k}\right)=6^x-5^y.$

What is the value of $x+y?$

Note: This problem is not original. It is adapted from a question posted on Math SE.

512 128 256 32 14 20 64 54