The lifetimes of the first 2$^{+}$ states in the neutron-deficient $^{76,78}$Sr isotopes were measured using a unique combination of the $gamma$-ray line-shape method and two-step nucleon exchange reactions at intermediate energies. The transition rates for the 2$^{+}$ states were determined to be $B$(E2;2$^{+}$$to 0^{+}$) = 2220(270) e$^{2}$fm$^{4}$ for $^{76}$Sr and 1800(250) e$^{2}$fm$^{4}$ for $^{78}$Sr, corresponding to large deformation of $beta_2$ = 0.45(3) for $^{76}$Sr and 0.40(3) for $^{78}$Sr. The present data provide experimental evidence for mutually enhanced collectivity that occurs at $N$ = $Z$ = 38. The systematic behavior of the excitation energies and $B$(E2) values indicates a signature of shape coexistence in $^{76}$Sr, characterizing $^{76}$Sr as one of most deformed nuclei with an unusually reduced $E$(4$^{+}$)/$E$(2$^{+}$) ratio.
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