  We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons
interact with each other via both a two-body interaction potential of the form
$a N^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction
potential of the form $-b N^{2\beta-2} W(N^\beta(x-y,x-z))$, where
$a\in\mathbb{R}$, $b,\alpha>0$, $0<\beta<1$, $U, W \geq 0$, and
$\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y)
\mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for
any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3\pi^2/2$ (the critical
strength of the 1D focusing quintic nonlinear Schr\"odinger equation) or for $a
\geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in
(0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system
exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic
NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to
0$ as $N \to\infty$, with the former convergence being slow enough and "not
faster" than the latter, we prove that the ground state of the system is fully
condensed on the (unique) solution to the quintic NLS equation. In the latter
case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for
small $\beta$ when $a > 0$ is fixed. Finally, we analyze the behavior of the
many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is
"faster" than the slow enough convergence $0<a_N \searrow 0$.
