When you use the line:

open (DAT, "+>>$file") or die "$!";
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You aren't just opening the file read/write, you're opening it read/write in *append* mode. That means the pointer is at the end of the file when you open it. So when you read it, you don't get anything. There are a few things you can try here to learn a little more. Here's a little debugger session I played with that might help you get this working correctly. There's also some other things I do differently here that you may wish to adopt as standard practices, such as using a lexical filehandle and 3-argument form of open:

  DB<1> open $fh, '+>>', 'test.dat'                                   
+                                                                     
+           
  DB<2> if ( eof $fh ) { print "Don't try to read, you're at the end!\
+n" }                                                                 
+           
Don't try to read, you're at the end!
  DB<3> seek $fh, 0, 0                                                
+                                                                     
+           
  DB<4> if ( not eof $fh ) { print "Read on, McDuff!\n" }             
+                                                                     
+           
Read on, McDuff!
  DB<5> @lines = <$fh>                                                
+                                                                     
+           
  DB<6> x @lines
  DB<7> chomp @lines                                                  
+                                                                     
+               
0  'First Line'
1  'Second Line'
2  'Last Line'
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Hope this helps, ~Jeff