A 18 mile by 57 mile plot of land is divided into \(1026\) counties of 1 mile by 1 mile squares. A man starts at the top right corner of the plot and travels a straight line path to the bottom left corner. In each county that he travels through, he spends \(-4\) times the sum of the amounts he spent in the previous two counties. He spends $1 in the first county and $2 in the second county. If he spends \(a\times 2^b\) dollars in the last county, where \(a\) is odd and \(b \geq 0\), find \(a+b\).
This problem is posed by Jesse Z.
Details and assumptions
Spending a negative amount of money is equivalent to receiving money. As an explicit example, he receives $12 in the third county.