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.

Hint:  Work backwards.  Writing a number greater than $112$ loses, so writing a number in the range $56-111$ wins.  Keep going.  
This is an impartial game, which you can use the Sprague-Grundy theorem to analyze.
