Jeremy wrote all the three-digit integers from $100$ to $999$ on a blackboard. Then Allison erased each of the
$2700$ digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every
1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$

Let's think step by step and output the final answer within \boxed{}.
user