Periodic Kaprekar Numbers

A n-digits Kaprekar number, in base 10, is an integer such as its square can be split in 2 parts that add up to this number.

For instance:
${324324324324}^2=\underbrace{105186267348}\underbrace{219138056976}\text{ and }$ $105186267348+219138056976=324324324324$

We say that a string $S$ is periodic if there is a substring $T$ of $S$ such as $S=T\dots T$ where the number of repetitions of $T$ is $\ge 2$

How many 68-digits periodic Kaprekar numbers are there?

Answer format: count,sum

Example : $8,41153193346$ // For 10-digits Kaprekar numbers
$(1111111111,2020202020,3888938889,4132841328,5000050000,5243952439,9756097560,9999999999)$

[My timing: < 1 sec]