There are two files in the home directory: passwords.old and passwords.new. The password for the next level is in passwords.new and is the only line that has been changed between the two files.
When I logged in, I saw two large files. Looking through them manually would be a nightmare because they contain many similar-looking strings.
In Linux, there is a perfect tool for this called diff. It compares two files line by line and shows exactly what is different between them.
Here is the command I used:
$ diff passwords.old passwords.new
The output showed a few lines, but specifically, it pointed out the line that was removed from the old file and the new line added to the current file. The line marked with a > (indicating it belongs to the second file, passwords.new) was the password I needed.
[SPOILER]
diff is the most efficient way to find changes between two versions of a file.diff results—specifically that < refers to the first file and > refers to the second.