Paper coffee filter

Question

Is filtering coffee through a paper filter on shabbat permitted. Is a paper coffee filter considered a keli she'meyudo lekach, i.e. not yadayim or a cloth, which are permitted as per masechet Shabbat. Is there even an issue of borrer when it comes to separating a solid (coffee grounds) from liquid (the remaining water). Is there a problem of Meraked. Is there a difference between shabbat and yom tov. Thank you very much for clarifying