For a truth teller, at least two of the three people next to him to the right are liars. For a lier, no more than one of the three people next to him to the right are liars. Bases on these two conditions we find that the only two possibilities are: [asy][asy] size(5cm); label("T",dir(0));label("T",dir(40));label("T",dir(160));label("T",dir(200));label("L",dir(80));label("L",dir(120));label("$\cdots$",dir(240));label("$\cdots$",dir(280));label("$\cdots$",dir(320));draw(1.25E+1.25N--1.25E+1.25S); label("T",dir(0)+2.5E);label("L",dir(40)+2.5E);label("T",dir(160)+2.5E);label("L",dir(200)+2.5E);label("T",dir(80)+2.5E);label("L",dir(120)+2.5E);label("$\cdots$",dir(240)+2.5E);label("$\cdots$",dir(280)+2.5E);label("$\cdots$",dir(320)+2.5E); [/asy][/asy] Both result in $1000$ truth tellers.