  It is essential to directly investigate the self-couplings of gauge bosons in
the Standard Model (SM) due to its non-Abelian nature, as these couplings play
a significant role in comprehending the gauge structure of the model. The
discrepancies between the Standard Model's expectations and the measured value
of gauge boson self-couplings would serve as strong evidence towards the
existence of new physics phenomena that extend beyond the Standard Model. Such
deviations could provide valuable insights into the nature of new physics and
potentially lead to a deeper understanding of fundamental particles and their
interactions. In this study, we examine the sensitivities of anomalous
couplings associated with dimension-8 operators that affect the $\gamma \gamma
\gamma \gamma$ and $Z \gamma \gamma \gamma$ quartic vertices. The study focuses
on the process $e^- \gamma \to e^-\gamma\gamma$ with the incoming photon under
Weizs\"acker-Williams approximation at the stage-3 scenerio of Compact Linear
Collider (CLIC) that is refer to a CoM energy of 3 TeV. Due to the CLIC
options, we take into account the both unpolarized and $\mp80\%$ polarized
electron beam with the related integrated luminosities of ${\cal L}=5, 4, 1$
$\rm ab^{-1}$ under the systematic uncertainties of $\delta_{sys}=0, 3, 5$.
Obtained sensitivities on the anomalous quartic gauge couplings (aQGCs) for the
process $e^- \gamma \to e^-\gamma\gamma$ at $\sqrt{s}= 3$ TeV and various
polarizations, are improved by a factor of 2-200 times better for the couplings
$f_{T,j}/\Lambda^4$ compared with the experimental results.
