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Go back to E4. You shouldnt be able to get in, if you can't go to 1 island and talk to Celio, complete the side quest where you help loreli of the E4 defeat Team Rocket and then once shes retured to the E4 go back to Cerulean Cave and you can go and catch the mysterious Mewtwo

You have to visit Celio on One Island. He will ask you to obtain the Ruby and Sapphire gems. This quest takes quite a long time and you will need to visit all seven of the Sevii Islands in addition to defeating Team Rocket once again.

The quest is pretty linear. Once you have given the gems to Celio, you should be able to access Cerulean Cave.

You beat the Elite 4 & you're Rival but he hasn't moved. Here's why, he moves when you give Celio(Bill's friend from One Island) a Sapphire.

To get the Sapphire, you must catch 60 or 65 PKMN, show Oak your Pokedex, & he'll update it for you, & your Tri-Pass to a Rainbow pass, now you can go on islands 1-7. Go to 4 & Rival will be there, he'll talk, & basically you have to Cut a door on Seven Island I believe(you'll know it has Braille on it), jump some holes in some order(idk), & you'll find a sapphire but this one guy will take it to 5 island(i think, the one w/ Rocket Warehouse), you will enter.

Beat some rockets & find the guy who took the Sapphire, beat him & take the sapphire to Celio & now his system can interact with Hoehn(idk how to spell it), & you can battle Mewtwo in his Cave.

Hint-If you have your masterball, unless you want Entei, Suicune or Raikou, use it on Mewtwo. If you don't use it, than get a strong PKMN with false swipe(like scyther), use it till he has 1 health, than try to paralyze him with thunder wave or put him to sleep, throw some balls at him & good luck catching him.

Go to professor Oak, he will give you a ticket and you will be able to go to some islands by using a boat. Do whatever you can on those islands and he will eventually move.

Well first you will have to visit prof. Oak,he will give you a ticket.use it to get to 1 island.talk to Celio and the guy by Cerulean City should move

Ok,I've caught Mewtwo with a luxury ball,so don't use the Master Ball.save it for the legendary dog(appears in viridian Route 2,level 50,it can flee).you beat the Elite Four,the go to dotted cave(6 island,I believe)and use Cut on the door.fall in hole,go up,down,left,right,(if thats not it read the brail writing and see how many letters are there)youll find the Sapphire,then a man will take it away and give you a code for team rockets base.oh,first you need the Ruby which is in mt.ember,idk if you have to battle the people or not.anyways,go to Five Isle Meadow,Team Rocket base,find your way through there,and give the saffire to Celio after escaping.I hope this helps,mewtwo is level 70 and does not want to stay in a ball,bring strong Pokemon,escape rope,and plenty of pokeballs of all sorts,save before battling.cereleun cave is a great place to level up and catch Pokemon,my sibling loves Pokemon,he evolves so many.well,I hope this post helped you!time to disco!!!!!