Consider the region constrained by $1\le x$ and $0\le y\le 1/x$.

Let $S_1$ be the largest square that can fit under the curve.
Let $S_2$ be the largest square that fits in the remaining area, and so on.
Let the index of $S_n$ be the pair $(\text{left},\text{below})$ indicating the number of squares to the left of $S_n$ and the number of squares below $S_n$.

The diagram shows some such squares labelled by number.
$S_2$ has one square to its left and none below, so the index of $S_2$ is $(1,0)$.
It can be seen that the index of $S_{32}$ is $(1,1)$ as is the index of $S_{50}$.
$50$ is the largest $n$ for which the index of $S_n$ is $(1,1)$.

What is the largest $n$ for which the index of $S_n$ is $(3,3)$?