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We propose a recurrence entanglement purification protocol to purify the three-qubit W-like state affected by amplitude damping, and we give an analytic expression for the protocol's purification range. By using the protocol, we illustrate the purification for an asymmetric W state and its generalized state. Nonlocal and maximal bipartite entanglements with corresponding probabilities can be obtained as byproducts in the course of the purification. What is important is that we also show the necessity for the single-qubit operations in the purification protocol.