What would have happened if... Min has died on Dumai's Wells

Rand stared at Min, shock and grief battling each other inside him, the Void had gone as soon as he saw Min's eyes, staring at him accusingly, lifelessly.
No! Not again! Shouted Lews Therin, but he hardly noticed the man. He formed the Void inside him, it was easy, the white-red fury the overwhelmed him left no emotions behind it. Only this vast emptiness he remembered from the day he killed Rahvin. He drew as much as he could, and more. Faster than ever before and more of saidin that was safe for him to draw.
He took it all and stopped, even with that amount of the power he could do nothing, Min was dead. The thought was thousands knife cuts, more. The void broke, and he felt tears on his cheeks. Once, on Tear he tried to heal a dead child, with Callandor he could draw more then he could with any other sa'sangreal save one, and even Callandor was useless against death.
The only other option he had wasn't one he could take, not even for Min's life. He had been offered that kind of option before, when he knelt near another woman he killed. But now he hadn't the madness to excuse his action. He killed Min when he tried to break free, an accident, but it wasn't a excuse for him. He wouldn't turn to the Dark One, not for any price. That was one oath he meant to keep, he swore it in an other life, but it was still held for him, so many ages after he voiced it.

The Void formed, and he drew saidin again, he meant to do exactly what he had done before, but no one died when he last died, now he meant that every Aes Sedai that had captured him would die, everyone who had helped them.
No! This was Lews Therin. I will not die again! The flood of saidin that rushed inside him stopped. Now Lews Therin fought to take saidin from his grasp; fought harder than he ever did before, and Rand didn’t have the strength to fight anymore.

It was Rand al'Thor who knelt beside the woman's corpse, but Lews Therin was the one who rose from the ground, Rand was shut up to a buzzing fly. Lews Therin used air to lift the woman - Min, that was her name? - he wove all the Five Powers and traveled. Making Min and himself part of saidin, he felt the taint he had to pass to reach to the True Source, felt it burning inside him. Into his soul and bones, but ignored it. The True Source turned the Wheel of Time, and there wasn't a faster way to travel, and now time was important then anything else. The madness waited for him, he knew that more then well enough, but he could subside it. For a little time at least.
He laid the woman on the floor on his - Rand's, was there any difference between them now? - rooms on Cairhien. And run to the wardrobe, tearing up the traps he had woven and inverted around it, he uncovered two statues, a male and a female holding a crystal globe and wearing a flowing robe.
The grimace he wore was only partly due to what happened, the woman's death and his own imprisonment, the taint on saidin was probably cause directly because of those two statues.
He took the male one, the only one of the pair he could use, the other was of no use to him without a female to draw saidar to aid him
Saidin turns the Wheel of Time, and with the ter'angreal in his hands he could draw enough to make the breaking of the world look like a walk in the park. He reached for the True Source through the sa'angreal, faintly aware of the huge replica that shined brighter than the sun not very far away from the city; even during the day, it light up the skies like a second sun. He sent it all to the corpse on the floor, kneeling to touch her face.

It was better that way, all the Five Powers to search, to seek, the Lord of the Grave was one of the names people gave the Dark One. But the Trollocs called him The Lord of Death, and with the sa'angreal in his grasp he could challenge the creature himself, and win, maybe. Her breasts started moving, eyes flickering, he could hear the blood starting to rush again in her veins, could see her heart start beating. The amount of the power he drew was beyond his wildest dreams, and he directed it all into the woman body, he could keep it alive as much as he wanted without the sa'angreal, as an empty shell, a body without a soul. Her eyes opened, not the soulless stare he half expected for, dead eyes while the heart still beat and the body lived, but a live stare, tears trickled down her cheeks, and his. He took a deep breath and was only half aware that it was his first in a long time, almost since the moment he saw that she was dead. Her eyes closed again, but she breathed, and he could feel that there was a soul inside the body. There was one thing he still had to do, he had to move quickly. He knew that he was walking on the edge of madness, on the edge of razor of saidin., but Rand would never do this no matter how much he knew he had to. He traveled again, Min would be alright, she was sleeping now. But there was other thing he had to heal, another wound he create, though this was a long time ago.

On the very top of the Dragonmount, the closest thing he had he could call a grave, he looked at the White Tower . They needed to learn what it meant to challenge the Lord of the Morning.

This time he didn't draw from saidin, he drew the taint. Sending it all to the White Tower , a trail of darkness as black as balefire was white. He could feel it evilness even as he build walls to stop it from harming him. He had no idea how much time passed, it could have been hours, or day or maybe just a heartbeat, but finally he could reach for the male half of the True Source and feel only saidin, the taint had gone. Sent it to the place that was created because of the taint, to a place that dedicated all its effort in finding and severing men that could touch saidin.
The place from which the order came to those who called themselves Aes Sedai and put him inside a box. He still held on to the power, and he couldn't hear the faintest sound from the Tower. He directed the taint to the White Tower alone. No one there would have survived this. Not unless they had time to ready themselves, which they didn't.

Lews Therin released the male half of the One Power. Clean as the female one. And retreated to the shadows in the mind of the man that became the Dragon Reborn. Rand al'Thor could move, could feel, he cursed Lews Therin, every curse he ever heard, he even passed to the Old Tongue without noticing once or twice. But the man knew what he had doing and Rand wouldn't have undone anything of the man's doing even if he could have.

He didn't trusted Lews Therin's way of traveling, he didn’t know how this worked and he couldn't trust what he didn't know. The gateway he opened into the utter blackness out of the pattern, on the other hand, was well known to him. He left the White Tower behind him, nothing would survive the blast of darkness Lews Therin used to clean saidin and avenge the Aes Sedai. Though the books in the library and the ter'angreal the Tower kept would survive. He couldn't have thought of a fate worse than this to the Aes Sedai. Especially to the Reds, to suffer what he had to suffer every time he had touch the True Source, although it was very quick for them. The gateway closed behind him and the platform - the ancient symbol of the Aes Sedai - moved through the darkness to his destiny, Cairhien, and Min. One thing Lews Therin left behind was that he couldn't run from duty nor love, the man put it in a different manner, of course, Rand didn't doubt that Lews Therin could make Mat blush like ten suns. He sat on the platform and tried to think about something else then Elayne, Aviendha and Min. Unfortunately, or maybe fortunately, it was one battle he was losing in.