There Is More Than One Way To Do It - one bizarre way is

#!/usr/bin/perl
{
  local $/ = "INPUT SEQUENCE=";
  <>; # discard first chunk
  while(<>) {
    chomp;
    open O, '>', /^(\d+)/ and $1 or die "$!\n";
    print O $/,$_;
  }
}
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update - oh. You want bunches of 1000 records in separate files?

#!/usr/bin/perl
{
  my $file = 'File00000';
  local $/ = "INPUT SEQUENCE=";
  <>;
  while(<>) {
    chomp;
    unless ($.-2 % 1000) {
        open O, '>', ++$file or die "cant write '$file': $!\n";
    }
    print O $/,$_;
  }
}
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update - a bit of explanation: setting the input record separator $/ (see perlvar) to the token right after the file boundary lets the diamond operator <> (or readline) read up to and including that token as a single line into $_.

With chomp we remove $/ from the end; it is added up front when outputting. The ++$file is a string increment; doing that we get the next file name (File00001, File00002, ... ).

Since the first "line" (consisting of the record separator only) isn't interesting, we do a <> before the loop. Next line is then number 2 ($. - see perlvar), and $.-2 % 1000 (modulo 1000) is 0 at line 2, 1002, 2002, ... , so we (re-)open the output filehandle at that line count, which does an implicit close. See open.

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