How about

while (($input = <STDIN>) < 7)
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?

And the reason it is exiting is that you are using if, not elsif. So this bit:

if($input == 6) {
print "You have chosen option 6\n";
  } else {
exit;
  }
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always exits unless you choose option 6. You could have found out how your program was working with the debugger: perl -d program.pl, then use 's' or 'n' to step through each line of your program.

dave hj~