After a lot of seemingly pointless tasks, you are finally at the base of Jupiter Lighthouse, one of the longest and most important dungeons in the game. Without the Hover Jade, although you could make some progress, you wouldn't actually be able to get anywhere.
The weeds don't conceal anything, and there's nothing up the ladder, so head inside. Although you are already inside Jupiter Lighthouse, you can't go anywhere because of a big, thick door blocking the way. And the switch is on the other side. Go left, and follow the path until you arrive back outside (you will be spending a surprising amount of time outside).
Go right following the cliff to reach a small patch of weeds which you can Cyclone to get a Mint. For the sake of keeping your party alive, it is probably a wise move to use this on Jenna, since she's going to be the one with the whole-party healing Psynergy that you will come to abuse later on. Also, you will want to set either Cyclone or Hover to a Psynergy Shortcut (keep the other one as Move).
Go left and use the Cyclone portal to go down into some caves. The patch of weeds to the south at the junction hides a Mad Plant which you can kill for another Mint. If you need it, the door to the north leads to another Psynergy room with a Psynergy Stone you can use. Anyway, keep going right to another door. Go down the ladder and use Hover on the glowing purple spot to float across to the ledge on the right-hand side. You can follow this path to a Cyclone portal, which you can use to enter the Lighthouse properly. Go left, Move the pillar onto the button, and use it to get through to the next room to the north.
Climbing the Lighthouse
This is one of the main rooms of the Lighthouse, but the right-hand door leads nowhere, so go through the left door. You can't do anything down the ladder, so just go up to the stairs.
For this rolling pillar puzzle, move the pillar nearest you upwards. Then go to the left of the bottom two horizontal logs, and push this one up as well. Go around the top and push the vertical pillar left. Push the right-bottom horizontal pillar up, then push the vertical pillar right, into the gap, and hop across to get the Erinyes Tunic. This is probably best equipped on Piers. Go out the bottom door.
Use the Cyclone portal back in this room to reach the high ledge, and enter the door. Go up and Move the pillar into the hole. Now go down, ignoring the Cyclone portal, and go outside via the door at the bottom. Outside, go around to the next door, and go up the stairs. Go up to the door on the left, and follow the path up, until you get to the pillars forming the Reveal symbol. Use Reveal and hop across. You'll notice there's another Reveal symbol being formed by the small white blocks, but you can't use them yet. Instead, head north through the door.
You'll be in a hopping block maze here - go down to the platform with the groove on it and slide down. Move the top pillar all the way to the left, and the bottom pillar one to the left. Push the next pillar along up to the top of its groove. Climb the ladder, and then use the chain of pillars by the stairs to hop around to the area enclosed by blocks and take the Meditation Rod from the chest. Equip this on Sheba, then go back along the chain of pillars and exit the room to the south (ignore the stairs up).
Outside, you'll probably walk onto the button by accident and you'll see that this causes the bridge along the front of the Lighthouse to open up. Go down to the bridge and shove the white block onto it. Then go up, hit the button, and watch with satisfaction as the block plummets off the Lighthouse. That was fun. Now go back inside, and head up the stairs at the top of the pillar room. Slide down the groove up here, ignore everything else, and go up the ladder. Go right, and down to the door.
Ignore everything else and go up the next set of stairs to the left. Head down to the next door. You're on the aerie? What the....
Reactivating the Hover Circuits
Sadly, there's a big stone plug blocking off the well. You can't squeeze the Jupiter Star underneath it, so go down through the door back into the Lighthouse by going to the left across the aerie. Head down the stairs and you'll be in another familiar room. Slide down the groove and approach the ladder in the middle.
"Holder of the star...Show the power of the Anemos!"
Pretty simple then. Climb the ladder and use Hover at the top of the pillar. There'll be a huge release of energy, and hey presto, all of the Hover circuits in the lighthouse will be functional again! This allows you to use the floating blocks to jump across to the right. Head up to the door to the right. Go down the ladder, and push the white block up onto the purple spot. This will open up a bridge. Now go down to the other purple spot on the circuit and use Hover yourself, and float across to the platform. Grab the Red Key, then slide down the groove.
Go up the ladder and use the bridge of blocks you made to get to the pillar room again. Head around the front of the Lighthouse, across the dropping bridge (don't worry, it won't fall) by taking the bottom door out, then go down the middle stairs in the pillar room.
Down here, you can use the floating white blocks as a Reveal symbol to hop across and collect a Psy Crystal from the chest. Now, go back to the left across the Reveal pillars that you came across earlier, and head back down along the path until you reach the Cyclone portal you didn't take before. Now that you have the Red Key, use Cyclone to carry yourself up a floor, open the door with the Red Key, and head to the next door. Back outside, you need to use the gaps between the blocks to get past the large Whirlwind statue trying to blow you back across to the main tower. Once you're past, go around and enter one of Jupiter Lighthouse's side towers.
The First Lifting Arrow
The chest to the left in here is a Mimic, which you can fight to get a Psy Crystal. Head right and through the door. Follow the path around to reach another rolling pillar room, with a currently inaccessible Blue Key in the middle. You can't do anything else, so go around the corner of the right-hand horizontal pillar and push it down. Go around it to the door at the top.
If you want to get both of the chests here, you'll need to do a bit of thinking. Firstly, go right across the cracked tile (you'll have to employ the same strategy here as you did with the cracked ground in Dehkan Plateau) and then down the first path to the bottom of the room. Go across the cracked tile down here and get 306 coins from the chest. Allow yourself to drop to the floor below, and just push your way back up to the stairs. Back up here, go the same way, but instead leave the room to the bottom. Although this resets the pillar, if you moved it, it also resets the cracked tiles. Move the pillar off the edge of the platform, then go back along the cracked tiles, around the left edge of the room and use the pillar to jump across and get the Mist Potion from the chest. Now drop down a floor again, and go back up the stairs. Follow the same route as before to get to the door at the bottom of the room. Follow the path to the next room.
Head straight up (you can't do anything else) and you'll be in a room full of cracked tiles. Go left, first of all, and use the Hover spot to get to the solid ground on the left. Move the pillar onto the button. Follow the path of cracked tiles back diagonally across the room to the Hover spot on the right, and use this to float over the really cracked tiles below you. Walk to the door at the bottom that's now open, and go up to the next level via the stairs (remember not to screw up and fall down the trap tile here). Outside, go up to the top of the side tower, and push the piece of Hover Circuit into place. The archer will fire an arrow into the bum of another statue on the aerie of Jupiter Lighthouse, and lift one side of the plug in the well up.
Now, head back down. Use the trap tile to fall the floor below, then follow the path of cracked tiles on the next floor up a second time, and take the lone cracked tile in the centre of the room all the way back down to the enclosure with the Blue Key. Once you've grabbed it, you can Retreat to the beginning of Jupiter Lighthouse.
Getting Past the Blue Door
Now that the Hover circuits are all working again, go back into the Lighthouse, all the way until you reach the room with the beam of swirling light that you encountered earlier (the tall room which only had one way out of it). Head back through the left door, and climb down the ladder. Move the block onto the active Hover spot, and use the reformed bridge of blocks to exit this room via the stairs to the right. Go along the outside of the next room to the large room again.
Shove the piece of Hover circuit off, and move it into the gap to close the circuit and reactivate the Hover spot. Float across to the platform using Hover, then use Cyclone on the portal to go up a level. Go through the door. Whilst in here, shove the pillar off to open another shortcut. Go outside via the door at the bottom right, and move the seemingly useless spike tile into the Hover circuit. It will conduct a lightning bolt, which will reactivate the circuit! Now head back down to the big room with the swirling light beam, and use Hover on the spot to float into it. You'll be carried up a level.
At the top, go right across the blocks, down the ladder, and move the other block into place on the Hover spot. Climb the ladder again, and use the bridge of blocks to get a Potion from the chest. Now head out the door to the bottom right. Go up to the next door. In here, do NOT use the Cyclone portal - it's one way and will lead to a place you've already been, and you'll waste PP using Hover and Cyclone to get back. Use the Blue Key on the door and head outside. Use Hover on the spot and the big whirlwind statue will blow you straight across to the next side tower. Go around and enter it. Ignore the stairs on the right, and enter the next large room through the door straight ahead.
The Second Lifting Arrow
Use Hover to float up to the platform. Up here, shove the Hover circuit-piece off, then slide down using the groove. Move it along into the right-most of the three holes and use the newly-available Hover spot to float up and get the next piece of the circuit. Shove it off, then push it into the hole nearby to access the exit. But wait! There's a Jupiter Djinni to get. Use the right Hover spot to float up to the Pound pillar and use it to launch all the circuit pieces out of their holes. Go back down, and move the corner piece into the left-most hole, and the straight piece into the right-most one. You can now float up using the left Hover spot. Save your game, then defeat Whorl to have it join you. Set it to Sheba. Now either go back down, float across, use the Pound pillar again, and solve the puzzle as you did before to activate the top-left Hover spot to Hover to the door, or just Hover from the already activated spot to just beyond the barrier that Whorl was sitting next to, and continue through the door.
Another tedious puzzle - use Pound on the pillar to eject the Hover circuit piece. This serves as a hopping-stone so you can go down and re-adjust the blocks. Move it so you can use it to jump across to the ladder. Now, if you want a Water of Life, which you may very well, you can solve this puzzle as outlined below.
The image shown to the left is the solution to get the Water of Life from the chest. The image to the right is the solution to reach the exit. At the end of each route, go back up using the ladder and move the Hover circuit back into its hole. Use the Pound pillar to eject the Hover circuit, so you can access the blocks if you went for the Water of Life. Once you're done, leave the room to the door at the bottom left.
Push the pillar onto the button to raise the bridge, then go up the stairs. Go around the maze of blocks in the next room, until you reach the stairs. This next room is a bit confusing, so I'll guide you through the steps you need to take. Firstly, go left to the left-hand wall. Go down, and then back to the right, weaving between the cracked tiles and blocks. Run straight past the Whirlwind statue (don't let it blow you across), and go across the semi-cracked tile underneath the large hole to the floor below. Go down to the Whirlwind statue and let it blow you across to the right-hand side of the room, and use the semi-cracked tile to get across, and hug the right-hand wall down to the staircase.
The next room up is FULL of Hover pads. This basically means that once you use Hover you can stay floating for a while. Go to the left first - run past the Whirlwind statues before they blow you across (you will want to stay as close to the cracked tiles on the left side as possible, without actually going onto them and losing Hover time). For the second statue, you will have to go onto the cracked tiles, then weave back around them to the Hover pads before you run out of Hover power. Land on the small ledge and get the Phaeton's Blade. This should be equipped to Piers. Now, slide back down and cheat all the statues by walking under them. Back at the front of the room, use Hover again and float across the cracked tiles, past the door, to the Hover gauntlet on the right-hand side.
This side is harder - the first statue is easy enough, but the second requires absolutely precise timing - you need to Hover over the cracked tiles to get past, but go back onto them too early, and the Whirlwind statue will catch you. Down you go. Go back too late, and you run out of Hover time and fall. Down you go. But eventually you'll get it right, just sneak past the Whirlwind, and you can float onto the ledge and up the stairs. Follow the long path to the outside of the second side tower.
Up here, move the Hover circuit piece into the hole and the second archer will fire an arrow into the bum of the second statue, and congratulations - the plug on the well has been lifted! Go ahead and Retreat back to the beginning of the Lighthouse.
Back Up to the Aerie
Before you go any further, give Piers all the Waters of Life and Mist Potions you have accumulated. For the purposes of what's coming up, if there is any better equipment available for Piers that isn't equipped to Felix, make sure Piers has it. You can afford for Jenna and Sheba to be a little weaker, but not Felix or Piers. Make sure that Felix has all the Venus Djinn, because you will need access to Revive in case something goes sour.
Go back into the Lighthouse. Make your way up as you did the first time, solving the pillar puzzle again on your way. Go by the Cyclone pillar that leads to the Red Key door that you opened before and outside through the door at the bottom left. If you haven't done the preparations, and you want to do them, make sure you do it before heading through, or it'll be too late.
Take a step outside, and guess who it is - Issac and co. climbing the Lighthouse to stop you. Unfortunately, they fell foul of the trap you so deftly disarmed before, and now Mia's stuck on a ledge and Garet is hanging by a hand. And the terrible two, Agatio and Karst, are here to screw everything up. Once the scene is over, you need to head up to the same level Agatio and Karst are on. Approach them, and after the scene, it'll be you and Piers. If you've forgotten the way, head through the door that you're near, and make your way back up to the pillar room from there. Use the groove, and then the ladder to reach the other side. Alex will show up and heal your party's HP and PP fully - not much of an apology for stabbing you in the back, but useful nonetheless.
Once you and Piers are by yourself, head up to the aerie the same way you did before (go past where Alex stood before). When you get out onto the aerie, save, but don't standby any Djinn. You will need every ounce of strength you can get. Approach Agatio and Karst again, and cast in the Jupiter Star. Uh oh! Betrayal time!
Boss Battle: Agatio and Karst
You should have seen this coming really - you annoyed them before, so now they're going to kill you. The reason why you got Revive and Waters of Life ready is because Agatio and Karst are a pretty powerful bunch, and they can bulldoze Piers down in a matter of seconds with his Fire weakness. But at the same time he has Waters of Life so that if Felix dies he can be resurrected.
Don't focus on doing any real damage for the first two turns - then, thankfully, Jenna will show up. Now you're prepared to start dishing out damage. Have Jenna use Healing/Cool Aura every turn since her Fire Psynergies don't hurt either Karst or Agatio that much. Piers should be dishing out damage with Diamond Berg or attack Djinni, and the same with Felix and Odyssey until you can summon. A turn after Jenna shows up, Sheba will come up onto the aerie, completing your party.
As for taking Agatio and Karst down, focus on Agatio - he's more deadly, and if you use summons particularly, Karst's lower HP count will diminish at the same time, so a few more attacks after Agatio goes down, Karst will follow. In terms of attacks, they both have their own collection of powerful moves.
Agatio has Rolling Fire, a small-ranged Fire attack, Meteor Blow, which is high damage to everyone, and Rising Dragon which does massive damage to one character. On top of this he has Cage, which shouldn't ever be a problem since you should be faster than him. Cage basically serves the same purpose as Felix's Petra, or the special effect of Ulysses; one-turn immobilisation. Karst has Death Scythe, an attack that carries a chance of instant death, and Djinnfest, a move that puts the first Djinni of every party member into recovery. Together, they also boast a fair arsenal of Fire Psynergies, including Healing Aura.
Also, it's a good thing to point out that this is the only battle in this game you can actually lose - as in, you won't get Game Over if you party dies. You do, however, get a hefty amount of coins and a new type of forgeable material, Dark Matter, for winning. After the scene, head down the elevator to the base of Jupiter Lighthouse, save, then head back to Contigo.