MATH4. Reverse a Balance Beta

Let $n$ be a positive integer. There are $4n$ weights with masses $1,2,\ldots,4n$, placed on two pans of a balance so that both pans have equal total mass.

In one move, a single weight is transferred from one pan to the other. A weight may be moved more than once. After every move, the absolute difference between the total masses on the two pans must be at most $T$.

Find, in terms of $n$, the smallest positive integer $T$ for which it is always possible to make every weight finish on the pan opposite to its starting pan, regardless of the initial balanced arrangement. Prove that your value of $T$ is both necessary and sufficient.