Multiplying game strategy Given the following game, what is the strategy to win?
Given $X,N\in \mathbb{N}$  such that $N>X$ and $N>1000$, two players play against each other. Each player multiply $X$ by $2$ or by $3$ by his own choice. The player who reach $N$ or above- wins.
I realized that if it's my turn and my opponent reached $\lceil \frac{N}{3} \rceil$ I win, so I tried to see how can I "make" him get there recursively, but nothing solid came to my mind so I'm pretty stuck. 
Any help would be appreciated. 

The best method of attack for this is probably to work backwards. So, you see that if the number given to you is above $\frac{N} 3$, you win. What numbers, less than this, can you give to your opponent such that they have to give you a number at least $\frac{N}3$? Well, since they have to multiply by at least two, if you give them some number between $\frac{N}6$ and $\frac{N}3$, you will win on your next turn. For what numbers is it possible for you to give your opponent such a number? Well, anything between $\frac{N}{18}$ and $\frac{N}{6}$ will suffice, since you can choose which move to do.
You can continue backwards to figure out which numbers you have a winning strategy for (i.e. how can you force your opponents move to be in the desired interval)? An important hint on seeing the general strategy is this:

No matter what your opponent does, you can always ensure that the number increases by a factor of $6$ between their turns.

