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There is ABSOLUTELY NO EVIDENCE that the four gospel books were written in the first century. There is no evidence for their existence before the mid to late second century. Some Christians "believe" they were written in the first century simply because they WANT TO BELIEVE they were written then.
The first gospel we have for Isu Chrestos, aka Jesus Christ, was written about 140 CE , and it was called the Euangelion. The Euangelion was later modified by the Christians and renamed the gospel of Luke. The gospels called Mark and Matthew were copied from "Luke". Later, Irenaeus felt there were some errors in Luke, Mark, and Matthew so he wrote a new gospel which became known as John. All four biblical gospels were created in the later half on the second century, NOT the later half of the first century, so there was plenty of time to take information out of the works of Josephus to create the gospel stories.
As for not believing in the existence of the biblical Jesus, you seem to have forgotten what happened in the early 4th century when there were more than 50 different gospel stories about the biblical Jesus character, but those choosing which gospel stories were going to be included in their new testament went through the different gospel stories deciding, "Nope, that Jesus did not exist, and that Jesus did not exist, and that Jesus did not exist, and that Jesus did not exist" until they decided all the Jesus characters did not exist except for the character created in the four gospel stories they decided to keep for their new testament.