Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number is now greater than Paula's by $5070$. What was the sum of their original numbers?

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