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We discuss proofs of two algorithms that sandwich $H$ between an upper-layer free-product factor of $F$ that contains $H$ and a lower-layer free-product factor of $F$ that is contained in $H$.\n  Richard Stong showed that the unique smallest-possible upper layer, denoted $\\operatorname{Cl}(H)$, is visible in the output of the polynomial-time cut-vertex algorithm of J. H. C. Whitehead. Stong's proof used bi-infinite paths in a Cayley tree and sub-surfaces of a three-manifold. 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