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Precision measurements constitute only one facet of a sound high-energy-physics project for the future. They need to be complemented with direct searches for new phenomena at an energy-frontier facility, to possibly answer fundamental questions that will unavoidably arise from these precision measurements.   Both the $\epem$ Higgs factories discussed in Section~\ref{sec:Higgs} are accompanied by high-energy upgrade projects. For the ILC, it is foreseen to double its length to reach a centre-of-mass energy of 500 GeV, and it is not yet excluded to extend it by another factor of two towards $\sqrt{s} =$ 1 TeV. Another linear collider project, CLIC~\cite{cite:CLICDR}, could also take over for the high-energy physics programme all the way to $\sqrt{s}=$ 3 TeV. The foreseen upgrade of TLEP is of another, unique, nature. It would consist in re-using the 80-to-100 km tunnel for a very-large-energy large hadron collider (VHE-LHC). If instrumented with magnets of 16 T, pp collisions would be produced at a centre-of-mass energy of 80 to 100 TeV. Typically, $\epem$ can pair-produce new particles with masses up to half the centre-of-mass energy, should they be either electrically charged or with a non-vanishing coupling to the Z. The reach of ILC500, ILC1000 and CLIC is therefore limited to particles lighter than 250, 500 and 1500 GeV/$c^2$, respectively. With the absence of new phenomena discovery at the LHC so far, air is therefore getting extremely thin for the ILC even in its hypothetical 1-TeV version. The next LHC run at 13 TeV, expected to start in 2015, should bring the final word in this respect. A discovery of a new particle lighter than 1.5 TeV/$c^2$ in the 13-TeV LHC data would probably rejuvenate the proposal of CLIC at $\sqrt{s} =$ 3 TeV. Neutral particles with sizable coupling to fermions -- like new gauge bosons, for example -- can also be produced at $\epem$ colliders as $s$-channel resonances all the way to the highest centre-of-energy, but those are already excluded up to several TeV/$c^2$ by the LHC data at 8 TeV, and have therefore reduced interest in this context.