[X] Plan Tricologe
-[X] Amu:
--[X] Make cake and chocolate milk, like we had with Kana the day we first met. Offer her some.
--[X] Ask for her side of the story. Listen to it.
--[X] Emphasize her self-worth, and the legitimacy of her feelings. (Not just the feelings she's suppressing, either.)
--[X] Tell Kana that it isn't wrong to want to not to have to fight or hurt people or save them or even be a good person. Tell her it also isn't wrong wanting to fight, hurt and save people. It's fine for someone to want both at once.
---[X] Show Kana telepathically the fight you had at Nikaidou's lab long ago. Amu, as Amulet Heart with Ran, wanted nothing more than to brain Nikaidou with a heavy rod for kidnapping her Charas. But Su wanted to help him instead, to heal and save him despite his sins. Despite differing mindsets, neither parts of Amu were "wrong".
--[X] Chara-change in turn with Ran and Su, displaying the proof that it is fine for someone to want to be 2 or even 3 or more different things at once: After all, you and Utau live it every day, and surely both of us examplify this being ok?
--[X] Remind Kana you didn't give up on Utau, despite everything she did, and you won't give up on Kana either. So if Kana doesn't want to give up on her mother too despite everything, that's also fine. Whatever she decides, you'll help and be with her to the best of your ability.
--[X] There may be a way to heal Yui, if she wants to try, but it will require Kana's willing help.
---[X] Show Kana how you retrieved your own fragments from the Road of Stars - Ran, Miki and Su - and offer to either lend her Dia's assistance or put her in touch with Tsukasa, who seems to have his own way of getting there, to try the same thing.
--[X] In the meantime, we can try to help you take care of Yui after you teach us how, if you think you need some time off without having to be so worried about her?
--[X] Let Naoto take the lead in the event of a fight