Induction from two linked segments with one half border and cuspidal reducibility

In this paper, we determine the composition series of the induced representation $\delta([\nu^{\frac{1}{2}}\rho,\nu^c\rho])\times \delta([\nu^{-a}\rho,\nu^b\rho]) \rtimes \sigma$ where $a, b, c \in \mathbb{Z}+\frac{1}{2}$ such that $\frac{1}{2}\leq a < b < c$, $\rho$ is an irreducible cuspidal unitary representation of a general linear group and $\sigma$ is an irreducible cuspidal representation of a classical group such that $\nu^\frac{1}{2}\rho\rtimes \sigma$ reduces.