  Dark energy might interact with cold dark matter in a direct,
nongravitational way. However, the usual interacting dark energy models (with
constant $w$) suffer from some catastrophic difficulties. For example, the
$Q\propto\rho_{\rm c}$ model leads to an early-time large-scale instability,
and the $Q\propto\rho_{\rm de}$ model gives rise to the future unphysical
result for cold dark matter density (in the case of a positive coupling). In
order to overcome these fatal flaws, we propose in this paper an interacting
dark energy model (with constant $w$) in which the interaction term is
carefully designed to realize that $Q\propto\rho_{\rm de}$ at the early times
and $Q\propto\rho_{\rm c}$ in the future, simultaneously solving the early-time
superhorizon instability and future unphysical $\rho_{\rm c}$ problems. The
concrete form of the interaction term in this model is $Q=3\beta H
\frac{\rho_{\rm{de}}\rho_{\rm{c}}}{\rho_{\rm{de}}+\rho_{\rm{c}}}$, where
$\beta$ is the dimensionless coupling constant. We show that this model is
actually equivalent to the decomposed new generalized Chaplygin gas (NGCG)
model, with the relation $\beta=-\alpha w$. We calculate the cosmological
perturbations in this model in a gauge-invariant way and show that the
cosmological perturbations are stable during the whole expansion history
provided that $\beta>0$. Furthermore, we use the Planck data in conjunction
with other astrophysical data to place stringent constraints on this model
(with eight parameters), and we find that indeed $\beta>0$ is supported by the
joint constraint at more than 1$\sigma$ level. The excellent theoretical
features and the support from observations all indicate that the decomposed
NGCG model deserves more attention and further investigation.
