Squares $R$, $S$, and $T$ have integer-coordinate vertices and bottom edges on the $x$-axis. Square $R$ has side length $8$ and lies to the right of the $y$-axis, with its left edge on the $y$-axis. Square $S$ has side length $6$ and lies to the left of the $y$-axis, with its right edge on the $y$-axis. Square $T$ has side length $4$, with its bottom edge from $(-3,0)$ to $(1,0)$. How many lattice points belong to exactly one of $R$, $S$, and $T$?

1. The squares contain $81$, $49$, and $25$ lattice points in $R$, $S$, and $T$, respectively.
2. The pairwise intersections contain $7$ points in $R\cap S$, $10$ in $R\cap T$, and $20$ in $S\cap T$. The triple intersection contains $5$ points.
3. By inclusion-exclusion, the numbers belonging to exactly one square are $81-7-10+5=69$ in $R$ only, $49-7-20+5=27$ in $S$ only, and $25-10-20+5=0$ in $T$ only.
4. Therefore, the answer is $69+27+0=\boxed{96}$.