You start with the number \(1\) on a piece of paper. You may perform two operations:
double up (multiply the number on your paper by \(2\), and erase the old number), increase by \(1\) (add \(1\) to the number on your paper, and erase the old number).
So for example, you may end up with the numbers \(1,2,3,6,\cdots\). Show that it is possible to obtain \(975\) starting from \(1\) in at most \(16\) operations.