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A Storm of Swords

Sandor and his “squire" are captured and bringed to Stoney Sept in the Riverlands. The Brotherhood without Banners takes them to Hollow Hill. Clegane is put on trial for various atrocities committed by Lannister soldiers, but he survives and is freed.

Arya is recognized by Harwin and Sansa. The two sisters reunite and remain under the "protection" of the outlaws. Because of this, the story takes a very different turn from here on.

For example, I don't think Arya would try to escape the Brotherhood so soon. As a result I don't think the Hound would be unable to kidnap the Stark girls again. However, the Hound could also decide to stay with the Brotherhood. He could plan to go to the Twins, introduce himself to Robb as Arya's savior, and ask him for a job.

So the Brotherhood proceeds as planned, they now have another valuable hostage and intend to take both girls to Lord Edmure's wedding and ransom them to their family. A group of men (like Lem, Harwin, Tom, etc) accompanies the sisters to Harroway to cross the Trident, but their journey is delayed because they find it flooded (like in Arya IX).

They reach the Twins just in time for the Red Wedding, and the outlaws manage to drag the Stark sisters away and save them. Arya and Sansa go through a complicated period of mourning but the fact of being together helps them. The two want to hold onto hope that perhaps their mother might have survived.

One night Arya has her first wolf dream in a long time: she sees Cat's body and drags it out of the river. In the morning Sansa suggests asking the men to go back and look for the woman, but Arya tells her that she's dead. As per canon Lord Beric, Thoros and the others come across the corpse and Dondarrion dies to resurrect her.

Lem's group continues their journey, this time they intend to take the girls to Lysa Arryn, but they discover that the mountain clans are bolder than ever and decide not to take the risk and return to the Riverlands.

The Stark sisters are getting impatient, Arya suggests that the two could run away and try to get to Winterfell alone. Sansa has to inform her sister that Winterfell was conquered by Theon months ago. She is devastated and abandons all plans. Lem's group returns to the Hollow Hill to discuss a new plan with Beric or perhaps to take more men as escorts before returning to the Vale.

Waiting for them in the hill, there isn't the lightning lord but Lady Stoneheart! Mother and daughters reunite.

A Feast for Crows