  Gauge boson self-couplings are exactly determined by the non-Abelian gauge
nature of the Standard Model (SM), thus precision measurements of these
couplings at the LHC provide an important opportunity to test the gauge
structure of the SM and the spontaneous symmetry breaking mechanism. It is a
common way to examine the physics of anomalous quartic gauge boson couplings
via effective Lagrangian method. In this work, we investigate the potential of
the process $pp\rightarrow p\gamma p\rightarrow p W Z q X$ to analyze anomalous
quartic $WWZ\gamma$ couplings by two different CP-violating and CP-conserving
effective Lagrangians at the LHC. We calculate $95\%$ confidence level limits
on the anomalous coupling parameters with various values of the integrated
luminosity. Our numerical results show that the best limits obtained on the
anomalous couplings $\frac{k_{0}^{W}}{\Lambda^{2}}$,
$\frac{k_{c}^{W}}{\Lambda^{2}}$, $\frac{k_{2}^{m}}{\Lambda^{2}}$ and
$\frac{a_{n}}{\Lambda^{2}}$ at $\sqrt{s}=14$ TeV and an integrated luminosity
of $L_{int}=100$ fb$^{-1}$ are $[-1.37;\, 1.37]\times 10^{-6}$ GeV$^{-2}$,
$[-1.88; \, 1.88]\times 10^{-6}$ GeV$^{-2}$, $[-6.55; \, 6.55]\times 10^{-7}$
GeV$^{-2}$ and $[-2.21;\,2.21]\times 10^{-6}$ GeV$^{-2}$, respectively. Thus,
$\gamma p$ mode of photon-induced reactions at the LHC highly improves the
sensitivity limits of the anomalous coupling parameters
$\frac{k_{0}^{W}}{\Lambda^{2}}$, $\frac{k_{c}^{W}}{\Lambda^{2}}$,
$\frac{k_{2}^{m}}{\Lambda^{2}}$ and $\frac{a_{n}}{\Lambda^{2}}$.
