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?

$\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$
1. Let Phillip's number be \( a \) and Paula's number be \( b \). According to the problem, Phillip's number is greater than Paula's number by 12. Therefore, we can write:
   \[
   a = b + 12
   \]

2. They each square their numbers and the sum of these new numbers is half of 169. Therefore:
   \[
   a^2 + b^2 = \frac{169}{2}
   \]

3. Substitute \( a = b + 12 \) into the equation:
   \[
   (b + 12)^2 + b^2 = \frac{169}{2}
   \]

4. Expand and simplify the equation:
   \[
   (b^2 + 24b + 144) + b^2 = \frac{169}{2}
   \]
   \[
   2b^2 + 24b + 144 = \frac{169}{2}
   \]

5. Multiply through by 2 to clear the fraction:
   \[
   4b^2 + 48b + 288 = 169
   \]

6. Rearrange the equation to standard quadratic form:
   \[
   4b^2 + 48b + 119 = 0
   \]

7. Factor the quadratic equation:
   \[
   (2b + 7)(2b + 17) = 0
   \]

8. Solve for \( b \):
   \[
   2b + 7 = 0 \quad \text{or} \quad 2b + 17 = 0
   \]
   \[
   b = -3.5 \quad \text{or} \quad b = -8.5
   \]

9. Since \( a = b + 12 \), calculate \( a \) for each \( b \):
   - If \( b = -3.5 \):
     \[
     a = -3.5 + 12 = 8.5
     \]
   - If \( b = -8.5 \):
     \[
     a = -8.5 + 12 = 3.5
     \]

10. Verify which pair satisfies the condition that Phillip's latest number is greater than Paula's by 5070 when squared again:
    - For \( b = -3.5 \) and \( a = 8.5 \):
      \[
      a^4 = (8.5)^4 = 5220.0625
      \]
      \[
      b^4 = (-3.5)^4 = 150.0625
      \]
      \[
      a^4 - b^4 = 5220.0625 - 150.0625 = 5070
      \]
    - For \( b = -8.5 \) and \( a = 3.5 \):
      \[
      a^4 = (3.5)^4 = 150.0625
      \]
      \[
      b^4 = (-8.5)^4 = 5220.0625
      \]
      \[
      b^4 - a^4 = 5220.0625 - 150.0625 = 5070
      \]

    Since both pairs satisfy the condition, we can use either pair. The sum of their original numbers is:
    \[
    a + b = 8.5 + (-3.5) = 5
    \]

The final answer is \(\boxed{5}\)