The James Webb Space Telescope (JWST) is becoming an essential tool in strong gravitational lensing studies. We propose to use the forthcoming JWST NIRCam and NIRSpec observations of galaxy cluster 1E 0657-56 (i.e. the Bullet cluster) as a testing ground for identifying multiply lensed galaxies and the development of the new lens models. We will evaluate the feasibility of automatic multiple-image identification methods. We will use the updated dataset to model the dark matter distribution in the Bullet cluster with unprecedented accuracy and use it to measure the displacement between dark matter and galaxies. Due to the extraordinary geometric and dynamical properties of the Bullet cluster, this will enable the most robust measurement of the dark matter self-interaction crosssection to date, thus helping to unveil the true nature of dark matter.