We present the analysis of a very high-magnification ($Asim 900$) microlensing event KMT-2019-BLG-1953. A single-lens single-source (1L1S) model appears to approximately delineate the observed light curve, but the residuals from the model exhibit small but obvious deviations in the peak region. A binary lens (2L1S) model with a mass ratio $qsim 2times 10^{-3}$ improves the fits by $Deltachi^2=181.8$, indicating that the lens possesses a planetary companion. From additional modeling by introducing an extra planetary lens component (3L1S model) and an extra source companion (2L2S model), it is found that the residuals from the 2L1S model further diminish, but claiming these interpretations is difficult due to the weak signals with $Deltachi^2=16.0$ and $13.5$ for the 3L1S and 2L2L models, respectively. From a Bayesian analysis, we estimate that the host of the planets has a mass of $M_{rm host}=0.31^{+0.37}_{-0.17}~M_odot$ and that the planetary system is located at a distance of $D_{rm L}=7.04^{+1.10}_{-1.33}~{rm kpc}$ toward the Galactic center. The mass of the securely detected planet is $M_{rm p}=0.64^{+0.76}_{-0.35}~M_{rm J}$. The signal of the potential second planet could have been confirmed if the peak of the light curve had been more densely observed by followup observations, and thus the event illustrates the need for intensive followup observations for very high-magnification events even in the current generation of high-cadence surveys.
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