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We report the results of X-ray (CXO) and radio (ATCA) observations of the pulsar wind nebula (PWN) powered by the young pulsar PSR J1016--5857, which we dub "the Goose" PWN. In both bands the images reveal a tail-like PWN morphology which can be attributed to pulsar's motion. By comparing archival and new CXO observations, we measure the pulsar's proper motion $\mu=28.8\pm7.3$ mas/yr, yielding a projected pulsar velocity $v \approx 440\pm110$ km/s (at d=3.2 kpc); its direction is consistent with the PWN shape. Radio emission from the PWN is polarized, with the magnetic field oriented along the pulsar tail. The radio tail connects to a larger radio structure (not seen in X-rays) which we interpret as a relic PWN (also known as a plerion). The spectral analysis of the CXO data shows that the PWN spectrum softens from $\Gamma=1.7$ to $\Gamma\approx2.3-2.5$ with increasing distance from the pulsar. The softening can be attributed to rapid synchrotron burn-off, which would explain the lack of X-ray emission from the older relic PWN. In addition to non-thermal PWN emission, we detected thermal emission from a hot plasma which we attribute to the host SNR. The radio PWN morphology and the proper motion of the pulsar suggest that the reverse shock passed through the pulsar's vicinity and pushed the PWN to one side., Comment: 13 pages, 10 figures; accepted for publication in ApJ