You can only do this quest after getting the Aibirds, so go get the two Spheres from Tomb of Aifread and Shizel's Castle first.
Fly north of Farlos to find a tower. Land and go in. Now, we'll do this by floors:
Ground floor, easy. Step on the buttons one after another to light them all.
First floor, memorise/write down the order the circles flash in, then repeat it by examining them.
Second floor, use your Ring to freeze the mouths blowing air, then ligth the candles with your other Ring. Stand still for a few seconds on the platform that appears to continue.
Third floor, light all the lights using your Ring. Shoot them in this order: Bottom-left, top-left, top-right, bottom-right, bottom-right, bottom-middle, top-middle.
Fourth floor, step on the panel and move around. When the ghostly panel touches a light, the light goes out.
Fifth floor, move around the coloured panels and light the rods. Use this map, and this guide:
Sixth floor, step on the pad, then step on these pads; far-left, up, right, right, up, up.
Examine the load point. Fight. Reid will fight alone. Get all the goodies after you win. The teleport will take you right to the bottom.
Now, go back to 57, 112 underwater in Celestia (remember this place?) and examine the sword for a totally awesome thing. I am so not telling you what. :P