Modify json with string in it

I have a json file like this:

{"arrays": ["a_string",    {"name": "foo","properties": [        {"type": "some_type1","url": "https://example.com","checksum": "d6fd580fe890b826250aa5d5a88bcdb28839cff4d447ba51536cda4ceae89c4e"        }      ]    },"another_string",    {"name": "bar","properties": [        {"type": "some_type2","url": "https://example.org","checksum": "d6fd580fe890b826250aa5d5a88bcdb28839cff4d447ba51536cda4ceae89c4e"        }      ]    }  ]}

I want to modify "a_string" to say "b_string" and the properties block of foo with something like jq '(.arrays[]| select(.name == "foo")).sources |= [{"type" : "sometypeB", "file" : "filename"}]'.

But select(.name == "foo") fails here due to the string not having any name, any idea how I can modify both?

The final json should look like:

{"arrays": ["b_string",    {"name": "foo","properties": [        {"type": "sometypeB","file": "filename"        }      ]    },"another_string",    {"name": "bar","properties": [        {"type": "some_type2","url": "https://example.org","checksum": "d6fd580fe890b826250aa5d5a88bcdb28839cff4d447ba51536cda4ceae89c4e"        }      ]    }  ]}