Problem of the Day #114

POTD April 1, 2024
Happy April Fools! This problem was inspired from China TST.

The numbers of sequences $(a_1,...,a_{100})$ whose terms are all nonnegative integers less than $97$ such that $97$ divides the sum of all the terms and the sum of the squares of all the terms can be written as $a^b$ where $\sqrt[n]{a}$ is not an integer for any $n\ge 2$. Find $a+b$.