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A TeV spectral break in the total flux of cosmic-ray electrons and positrons (CREs) at which the spectral power index softens from $\sim 3$ to $\sim 4$ has been observed by H.E.S.S. and recently confirmed by DAMPE with a high significance of $6.6~σ$. Such an observation is apparently inconsistent with the data from other experiments such as Fermi-LAT, AMS-02 and CALET. We perform a global analysis to the latest CRE data including Fermi-LAT, AMS-02, CALET, DAMPE and H.E.S.S. with energy scale uncertainties taken into account to improve the consistency between the data sets. The fit result strongly favors the existence of the break at $\sim 1$ TeV with an even higher statistical significance of $13.3~σ$. In view of the tentative CRE break, we revisit a number of models of nearby sources, such as a single generic Pulsar Wind Nebula (PWN), known multiple PWNe from the ATNF catalog, and their combinations with either an additional Dark Matter (DM) component or a Supernova Remnant (SNR). We show that the CRE break at $\sim 1$ TeV, together with the known CR positron excess points towards the possibility that the nearby sources should be highly charge asymmetric. Among the models under consideration, the one with a PWN plus SNR is most favored by the current data. The favoured distance and age of the PWN and SNR sources are both within $0.6$ kpc and around $10^{5}$ yr respectively. Possible candidate sources include PSR J0954-5430, Vela and Monogem ring, etc. We find that for the models under consideration, the additional DM component is either unnecessary, or predicts too much photons in tension with the H.E.S.S. data of $γ$-rays from the direction of the Galactic Center. We also show that the current measurement of the anisotropies in the arrival direction of the CRE can be useful in determining the property of the sources.