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Exit south, and you will be outside.
Go to the dome to the south.
Note: If you are playing on an emulator and can't see anything in the dome except for fog, you are probably using an old version of the emulator. Look online for the most recent version of the emulator. If that doesn't work, change your emulator's video mode, trying different modes until you can see properly.
Buy better equipment. Heal up in the Enertron, then exit the dome and go to the ruins to the north.
In Lab 16, don't touch the rats or they will steal from you.
First, go right and up for a chest (Berserker). Then go left and up, and follow the path. There will be another chest (Lode Sword). Hug the top to avoid getting attacked. Keep following the path, and go into the little crevice where the rat is running to get another chest (Lode Bow). Hug the bottom of the next area to avoid fighting, and try to maneuver above the next monsters to avoid them too.
Soon you'll get to the next area. Hug the bottom and go straight down and a little left to find a chest (Ether), then head right. You can avoid any fights by going up to where that red monster is and hugging the wall below it. If you fight Shadows, use Crono's Slash, Lucca's Flame Toss, or Fire Whirl. Keep going, hugging the left wall near the red monster, and soon you'll be out of there.
Head to Arris Dome. Buy stuff from the kid behind the boxes, use the Enertron, and save. Then go down the ladder. From there go up the left ladder. Keep going and eventually you'll be attacked.
First beat the bits, then attack the Guardian. If the bits are alive when you attack the Guardian, they'll counterattack. When the countdown reaches 0, stop attacking the Guardian. The bits will be revived. Kill them again, then go for the Guardian. Just use normal attacks. Keep your HP above 50.
Exit north. Check on the guy to read his note, then get the chest (Mid Ether). Leave.
Chase the rat by holding down B, then pressing A when you're next to it. If you don't get it, just leave and re-enter to try again. You'll get a password.
Now if you want you can go back up to the Enertron to heal up and save, or you can just head into the next area.
Go up to the sparkling console and enter the password. If you're using an emulator and you can't get the password to work, set L, R, and A to the same button on the keyboard, and press that key. When you've gotten it to work, the passage to the door will open. Go through.
Go up past the rats and robot for a chest (Mid Ether). Then go back down and take the left stairs. Follow the path.
Go up the stairs behind the robot, then take the stairs to the left, and go into the room. Say okay when asked.
After that's done with, go back up to Doan and the others. Then use the Enertron and save.
Head outside. There's a sewer entrance to your right, but you can skip that for now and come back later. If you DO go through, you can meet Belthasar, who says some weird stuff. It's entirely up to you. If you've already played through the game before, it's interesting to talk to Belthasar, but if this is your first run through the game, I say skip it.
Go up to Lab 32. Get the chest (Mid Tonic), then go up. Check on the car thingy or try to exit right. Then check on the car again to start the race.
The trick to winning is to use a B button boost when you're close to the finish line. Check how close you are to the finish line at the bottom of the screen. It also helps to try to get in front of Johnny so you bounce forward off of his car.
Or, you can just lose the race and exit right. You'll have to fight some monsters, but you'll get a chest.
If you win, exit left and fight some monsters to get the chest. Then, talk to Johnny and you can keep score of your race times if you race again! There may be a bug with it, because in order for my high scores to be recorded, I had to switch modes and back. Finally, exit Lab 32.
Skip the factory for now and go to the dome to the south. Fight the monsters, use the Enertron, and go to the top of the dome. Check the robot (default: Robo).
Now you have to choose who to leave behind. I say to leave Lucca behind so you have Marle's healing magic. You can always go back and switch party members later.