How does the Entity framework look up the right ADO.NET provider?

I am working with Entity Framework and SQLite. The EF provider for SQLite does not configure itself properly. In specific, the EF provider for SQLite is looking for the ADO.NET provider "System.Data.SQLite.SQLiteFactory".

The comment provided by SQLite team suggests that they want to avoid precisely this.

<!--
    NOTE: The extra "remove" element below is to prevent the design-time
          support components within EF6 from selecting the legacy ADO.NET
          provider for SQLite (i.e. the one without any EF6 support).  It
          appears to only consider the first ADO.NET provider in the list
          within the resulting "app.config" or "web.config" file.
-->

To troubleshoot this problem, I need to understand how Entity framework figures out the ADO.NET provider to load for a specific EF provider. Or how is the mapping between the EF provider and ADO.NET provider done behind the scenes? Refer to the diagram below for a better understanding of what I am trying to figure out.