LINE: while (<LOGIN>) {
    chomp;
    if (s!\\$!!) {
        $_ .= <LOGIN>;
        redo LINE;
    }
    # At this point, we have gathered all the continuation lines into 1 line.
    # Now do the rest of the processing.
    next unless /PATH/;
    # here you have your PATH line, and can do whatever you need with it.
}