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My understanding of the current situation is:
There is still uncertainty about the composiiton of the atmosphere of the early Earth. It may have had a composition that would rule out Urey-Miller synthesis of amino acids. More likely, though the atmosphere did permit Urey-Miller synthesis and it was a significant source of amino acids.
There has been a huge amount of work on later stages. One of the major discoveries was that RNA could work as a catalyst as well as a template for replication. This solved a major problem with the early stages of life because the same substance could perform two roles crucial to life, and it is now thought that DNA-based life developed from earlier RNA-based life.
There's a lot more but I'll leave that to people who are better informed