  Cluster-scale strong lensing is a powerful tool for exploring the properties
of dark matter and constraining cosmological models. However, due to the
complex parameter space, pixelized strong lens modeling in galaxy clusters is
computationally expensive, leading to the point-source approximation of
strongly lensed extended images, potentially introducing systematic biases.
Herein, as the first paper of the ClUsteR strong Lens modelIng for the
Next-Generation observations (CURLING) program, we use lensing ray-tracing
simulations to quantify the biases and uncertainties arising from the
point-like image approximation for JWST-like observations. Our results indicate
that the approximation works well for reconstructing the total cluster mass
distribution, but can bias the magnification measurements near critical curves
and the constraints on the cosmological parameters, the total matter density of
the Universe $\Omega_{\rm m}$, and dark energy equation of state parameter $w$.
To mitigate the biases, we propose incorporating the extended surface
brightness distribution of lensed sources into the modeling. This approach
reduces the bias in magnification from 46.2 per cent to 0.09 per cent for $\mu
\sim 1000$. Furthermore, the median values of cosmological parameters align
more closely with the fiducial model. In addition to the improved accuracy, we
also demonstrate that the constraining power can be substantially enhanced. In
conclusion, it is necessary to model cluster-scale strong lenses with pixelized
multiple images, especially for estimating the intrinsic luminosity of highly
magnified sources and accurate cosmography in the era of high-precision
observations.
