Fill a $2\times n$ grid with each integer from $1$ to $2n$ such that every cell with a neighboring cell above or to the left of it has a greater number in it than those neighbors.

If the number of ways to fill the grid this way is $f(n),$ then what is $\displaystyle\lim_{n\to \infty}\frac{f(n)}{f(n-1)}?$

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