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[References: 81 tit.]

Recently, the ATOMKI experiment has reported new evidence for the excess of e+e-e+e− events with a mass ~17 MeV in the nuclear transitions of 44He, that they previously observed in measurements with 88Be. These observations could be explained by the existence of a new vector X17X17 boson. So far, the search for the decay X17->e+e-X17->e+e- with the NA64 experiment at the CERN SPS gave negative results. Here, we present a new technique that could be implemented in NA64 aiming to improve the sensitivity and to cover the remaining X17X17 parameter space. If a signal-like event is detected, an unambiguous observation is achieved by reconstructing the invariant mass of the X17X17 decay with the proposed method. To reach this goal an optimization of the X17X17 production target, as well as an efficient and accurate reconstruction of two close decay tracks, is required. A dedicated analysis of the available experimental data making use of the trackers information is presented. This method provides independent confirmation of the NA64 published results, validating the tracking procedure. The detailed Monte Carlo study of the proposed setup and the background estimate show that the goal of the proposed search is feasible.