I think you were traveling the right path with your first attempt there. But I'd like to invite you to rethink how you're collecting the information that you plan to process.

my @input = <FILEin>;

Collecting everything all at once. It's not necessary to consume the whole thing, since each part does not rely on another.

I think if you were to

while( my $line = <FILEin>)  {

That would make it clearer to see a solution.

Maybe apply a switch to let you know if this is the first line, which requires opening a new destination and writing to it, or a second line, which requires writing to the destination and then closing it.

my $first_line_indicator=1;
my $outfile_counter=0;
while (my $line = <FILEin>){
  if ($first_line_indicator==1){
    $outfile_counter++;

    # open new write destination
    # write $line;

    $first_line_indicator=0;
  }else{
    # write $line;
    #close write destination

    $first_line_indicator=1;
  }
}
[download]

I hope that points you in the right direction.