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In this paper, we explore the potential of the LHC to measure the rate of $\mathrm{p}\mathrm{p}\rightarrow \mathrm{p}~ WWγ~\mathrm{p}$ process, also to probe the new effective couplings contributing to the $WWγ$ and $WWγγ$ vertices. The analysis is performed at the $\sqrt{s}=13$ TeV, in the di-leptonic decay channel, and assuming 300 $fb^{-1}$ integrated luminosity (IL). In addition to the presence of two opposite sign leptons, a photon, and missing energy, the distinctive signature of this process is the presence of two intact protons flying few millimeters from the initial beam direction in both sides of interaction points which suppress the background process effectively. To exploit this feature of signal we benefit from forward detectors (FDs) placed about 200 meters from the interaction point to register the kinematics of tagged protons. In order to overcome the major sources of backgrounds, we introduced three categories of selection cuts dealing with objects that strike the central detector, protons hitting the FDs, and correlations of central objects and protons, respectively. We also evaluate the probability of pile-up protons to be tagged in the FDs as a function of the mean number of pile-up. Then the sensitivity of the LHC to observe this process and constraints on multi-boson effective couplings are extracted. The obtained expected limits show very good improvements for dimension-8 quartic couplings and competitive bounds on dimension-6 anomalous triple couplings w.r.t the current experimental limits. Therefore, we propose this process to the LHC experiments as a sensitive and complementary channel to study the multi-gauge boson couplings.