Because the traditional observable of charge-dependent azimuthal correlator $gamma$ contains both contributions from the chiral magnetic effect (CME) and its background, a new observable of $R_{Psi_{m}}$ has been recently proposed which is expected to be able to distinguish the CME from the background. In this study, we apply two methods to calculate $R_{Psi_{m}}$ using a multiphase transport model without or with introducing a percentage of CME-induced charge separation. We demonstrate that the shape of final $R_{Psi_{2}}$ distribution is flat for the case without the CME, but concave for that with an amount of the CME, because the initial CME signal survives from strong final state interactions. By comparing the responses of $R_{Psi_{2}}$ and $gamma$ to the strength of the initial CME, we observe that two observables show different nonlinear sensitivities to the CME. We find that the shape of $R_{Psi_{2}}$ has an advantage in measuring a small amount of the CME, although it requires large event statistics.
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