  The Gamow-Teller strength distribution covering the entire $\beta$-decay
window, up to 10.312(4) MeV, of $^{80g+m}$Ga was measured for the first time in
photo-fission of UC$_x$ induced by 50 MeV electron beam. The new data show
significant enhancement in the high-energy region with a jump-structure.
Simultaneously, the $\gamma$ de-exciting behavior of $\beta$-populated states
presents a competition between de-excitation to 2$_1^+$ [$\beta_2$ = 0.155(9)]
and to 2$_2^+$ [$\beta_2$ = 0.053$_{0.009}^{0.008}$)] in $^{80}$Ge. Based on
these facts and combined with a realistic shell model calculation and
systematic analysis of logft ratio between precursor $\beta$-decay to 2$_2^+$
and to 2$_1^+$ of Ga isotopes, we conclude that these phenomena evidence
simultaneous impacts of nuclear shell structure and collectivity on B(GT) and
its distribution and, therefore, the half-life of the precursor. These data
prove that the nucleus as a multi-nucleon correlated quantum system reacts as a
whole when $\beta$-decay occurs in contrast to simple single-particle
excitation. Additionally, the comparison with the theoretical results evidence
how challenging is the description of the experimental data obtained, and
render this experimental outcome a sound test for the theoretical models.
