  The Hubble constant $H_0$ tension has emerged as the most serious crisis in
modern cosmology, potentially indicating that the $\Lambda$CDM model may not
describe our universe accurately. In this paper, we establish a new,
cosmological-model-independent method to study the cosmic background dynamics.
Using the latest Pantheon+ Type Ia supernova (SN Ia) sample and the
model-independent SN Ia sample (P+1690), we derive values for the luminosity
distance, the Hubble parameter, and the deceleration parameter at five
different redshift points ranging from 0.12 to 0.52. Our analysis shows that
results obtained from the Pantheon+ sample align with the predictions of the
$\Lambda$CDM model within 2$\sigma$ confidence level (CL), while those obtained
from the P+1690 sample exhibit deviations of about $2\sim3\sigma$ CL.
Furthermore, we explore the equation of state (EoS) of dark energy and find
that while the EoS values from the Pantheon+ sample remain consistent with $-1$
within 2$\sigma$ CL, the P+1690 sample does not conform to this standard. These
findings remain unchanged after the inclusion of the Hubble parameter
measurements in our analysis. Our results indicate that the $\Lambda$CDM model
remains compatible with the Pantheon+ SN Ia and the Hubble parameter
measurements at 2$\sigma$ CL.
