Problem 24 (2005 AMC 8)

A certain calculator has only two keys $[+1]$ and $[\times2]$. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "$9$" and you pressed $[+1]$ it would display "$10$." If you then pressed $[\times2]$ it would display "$20$." Starting with the display "$1$," what is the fewest number of keystrokes you would need to reach "$200$"?