Numbers with Additude: Middle Secondary Mathematics Competition Question

Every number has additude! Take a natural number (in base 10) and add together its digits; this gives us its digital sum. Continue this process to obtain a sequence of digital sums that terminates with a single-digit number; this is known as the digital root of the original number (and, of course, the digital root of all the numbers in the sequence). Now, the number of iterations needed to reach the digital root is to be called the additude of the starting number.

For example, starting with the number 5678 we obtain the sequence (5678, 26, 8). So the number 5678 has a digital root of 8 and an additude of 2. The number 8 has a digital root of 8 and an additude of 0.

Find the smallest possible number that has a digital root of 7 and an additude of 3.