15. Given $k$ as a positive integer, the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=3, a_{n+1}=\left(3^{\frac{2}{2 k-1}}-1\right) S_{n}+3\left(n \in \mathbf{Z}_{+}\right) \text {, }
$$

where $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$.
Let $b_{n}=\frac{1}{n} \log _{3} a_{1} a_{2} \cdots a_{n}\left(n \in \mathbf{Z}_{+}\right)$, and define
$$
T_{k}=\sum_{i=1}^{2 k}\left|b_{i}-\frac{3}{2}\right| \text {. }
$$

If $T_{k} \in \mathbf{Z}_{+}$, find all possible values of $k$.