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Continuing the investigations started in the recent work [12] on semi-global controllability and stabilization of the (1+1)-dimensional wave maps equation with spatial domain and target , where semi-global refers to the -energy bound, we prove global exact controllability of the same system for and show that the -energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in [12] is sharp. Lastly, the global exact controllability for -target within minimum time is discussed.