Problem D

Teacher Road Trip 2

To help pass time on a long bus ride, a group of math teachers created a sequence of numbers, with each teacher saying one term in the sequence. The first and second teachers each said a non-negative integer, and every teacher after that said the sum of all of the previous terms in the sequence.

For example, if the first teacher said the number \(2\) and the second teacher said the number \(8\), then

the third teacher would say the sum of the first and second terms, which is \(2+8=10\), and

the fourth teacher would say the sum of the first, second, and third terms, which is \(2+8+10=20\).

How many possible sequences could the teachers have said if the first teacher said the number \(3\) and another teacher said the number \(3072\)?