Solution:

Answer: $137$

Suppose that $a$ heads and $b$ tails are flipped. Jeff's number at the end is $4 + 8a - 3b$. Note that the operations which Steph applies are inverses of each other, and as a result it is not difficult to check by induction that her final number is simply $1 + 3 \cdot 2^{a-b}$.

We now have $4 + 8a - 3b = 1 + 3 \cdot 2^{a-b}$. Letting $n = a-b$, we see that $2^n - n - 1$ must be divisible by $5$, so that $a$ is an integer. In particular, $n$ is a positive integer. Furthermore, we have $1 + 3 \cdot 2^n < 2012$, so that $n \leq 9$. We see that the only possibility is for $n = 7 = a-b$, and thus $4 + 8a - 3b = 385$. Solving, we get $a = 72$, $b = 65$, so our answer is $72 + 65 = 137$.