A Game Involving Multiplying Numbers By An Amount From $2$ to $9$ Sean and Lila are playing a game on a blackboard. Sean starts by writing the number $1$. Then, in alternating turns (starting with Lila), each player multiplies the current number by any number from $2$ to $9$ (inclusive) and writes the new number on the board. The first player to write a number larger than $1000$ wins. A sample game might be (each number is preceded by "S" or "L" to indicate who wrote it):
$$\text{S1, L3, S12, L60, S420, L1260; Lila wins.}$$
Which player should win, and why?

This feels like a logic problem...  I don't know who should win and why!  I'm stuck.  Solutions are greatly appreciated.  Thank you.

Forget about names. the player who reaches $112$ first loses, since the other person multiplies by $9$ to win.
The player who reaches the range $56-111$ first wins, for the other person must at least double $56$, leading to $112$.
The player who reaches the range $7-56$ first loses, since the other player will be able to reach $56$ by multiplying by $9$.
Thus, all Lila needs to do is to take a number between three and seven, and that will force Sean to at least double this number to something more than $7$. Then Lila can follow the above algorithm to win.
Example:
S$1$ 
L$4$ (Lila plays a four)
S$16$ (Sean must at least double the number, he decides to quadruple it, and he can't avoid the range $7-56$).
L$80$ (Lila plays it so that she lands in the range $56-111$. If she made the mistake of multiplying by $9$ and getting $144$ for example then she would lose).
S$160$ (Resigned to his fate, Sean doubles the number hoping for an earthquake or tsunami)
L$1440$ (The natural disaster never comes, only one winner)
As an exercise, suppose that the first person to reach $1000$ loses. See who has a winning strategy in this game.
