  We propose methods to reconstruct particle distributions with and without
considering initial volume fluctuations. This approach enables us to correct
for detector efficiencies and initial volume fluctuations simultaneously. Our
study suggests such a tool could investigate the possible bimodal structure of
net-proton distribution in Au+Au collisions at $\sqrt{s_{\rm NN}}=$7.7 GeV a
signature of first-order phase transition and critical point
[arXiv:1804.04463,arXiv:1811.04456].
