  We study phase-transitions of the ferromagnetic $q$-state Potts chain with
random nearest-neighbour couplings having a variance $\Delta^2$ and with
homogeneous long-range interactions, which decay with the distance as a power
$r^{-(1+\sigma)}$, $\sigma>0$. In the large-$q$ limit the free-energy of random
samples of length $L \le 2048$ is calculated exactly by a combinatorial
optimization algorithm. The phase-transition stays first-order for $\sigma <
\sigma_c(\Delta) \le 0.5$, while the correlation length becomes divergent at
the transition point for $\sigma_c(\Delta) < \sigma < 1$. In the latter regime
the average magnetization is continuous for small enough $\Delta$, but for
larger $\Delta$ it is discontinuous at the transition point, thus the
phase-transition is of mixed order.
