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The path towards the choice of the most appropriate machine(s) to analyse these new phenomena may be guided by historical precedents, which reveal the important r\^oles played by lower-energy precision measurements that may often  establish roadmaps for future discoveries with higher-energy machines. In the late 1970's, precision measurements of neutral currents led to the prediction of the existence of the W and Z bosons, as well as the values of their masses. The W and Z were then discovered in the early 1980's at the CERN S${\rm p\bar p}$S collider with masses in the range predicted. Subsequently, the CERN LEP $\epem$ collider measured the properties of the Z and W bosons with high precision in the 1990's. These precise measurements led to the prediction of the top-quark mass, which was discovered at the FNAL Tevatron with the predicted mass. The measurement of $m_{\rm top}$, together with the precise measurement of the W mass at the FNAL Tevatron in the past decade, led in turn to an accurate prediction for the mass of the Higgs boson, which was recently discovered at the CERN LHC within the predicted mass range. The details of the optimal strategy for the next large facility after the LHC can only be finalized once the results of the LHC run at 13-14 TeV are known, but an emerging consensus is building up around the following general lines. A first step in the strategy to look beyond the LHC findings would require a facility that would measure the Z, W, top-quark and Higgs-boson properties with sufficient accuracy to provide sensitivity to new physics at a much higher energy scale. The strategy would then include a second step that would aim at discovering this new physics directly, via access to a much larger centre-of-mass energy.