A method for enrichment of at least one target nucleic acid in a library of nucleic acids, the method comprising: hybridizing a first oligonucleotide to a target nucleic acid in a library of nucleic acids, each of the nucleic acids in the library of nucleic acids having a first end comprising a first adapter and a second end comprising a second adapter; extending the hybridized first oligonucleotide with a first polymerase, thereby producing a first primer extension complex comprising the target nucleic acid and the extended first oligonucleotide; capturing the first primer extension complex; enriching the first primer extension complex relative to the library of nucleic acids; hybridizing a second oligonucleotide to the target nucleic acid; extending the hybridized second oligonucleotide with a second polymerase, thereby producing a second primer extension complex comprising the target nucleic acid and the extended second oligonucleotide, thereby liberating the extended first oligonucleotide from the first primer extension complex; and amplifying the target nucleic acid with a third polymerase, a first amplification primer, and a second amplification primer, the first amplification primer having a 3' end complementary to the first adapter and the second amplification primer having a 3' end complementary to the second adapter, wherein at least one uracil is incorporated into at least one of: the extended first oligonucleotide in the first primer extension complex, and the extended second oligonucleotide in the second primer extension complex, thereby forming a uracil-containing oligonucleotide product, or wherein the first adapter and the second adapter comprise at least one uracil. hybridizing a first oligonucleotide to a target nucleic acid in a library of nucleic acids, each of the nucleic acids in the library of nucleic acids having a first end comprising a first adapter and a second end comprising a second adapter; extending the hybridized first oligonucleotide with a first polymerase, thereby producing a first primer extension complex comprising the target nucleic acid and the extended first oligonucleotide; capturing the first primer extension complex; enriching the first primer extension complex relative to the library of nucleic acids; hybridizing a second oligonucleotide to the target nucleic acid; extending the hybridized second oligonucleotide with a second polymerase, thereby producing a second primer extension complex comprising the target nucleic acid and the extended second oligonucleotide, thereby liberating the extended first oligonucleotide from the first primer extension complex; and amplifying the target nucleic acid with a third polymerase, a first amplification primer, and a second amplification primer, the first amplification primer having a 3' end complementary to the first adapter and the second amplification primer having a 3' end complementary to the second adapter, wherein at least one uracil is incorporated into at least one of: the extended first oligonucleotide in the first primer extension complex, and the extended second oligonucleotide in the second primer extension complex, thereby forming a uracil-containing oligonucleotide product, or wherein the first adapter and the second adapter comprise at least one uracil. the extended first oligonucleotide in the first primer extension complex, and the extended second oligonucleotide in the second primer extension complex, thereby forming a uracil-containing oligonucleotide product, or wherein the first adapter and the second adapter comprise at least one uracil.