She could start by being honest with Percy about her feelings. Just come clean and say why she wants to get back together. For example, she might say she realized she can't live without him. This straightforward approach might help Percy see that she is sincere.
Well, one possibility is that Annabeth realizes she made a mistake in their break - up. Maybe she starts by trying to be around Percy more often. For example, she could join him on his quests or training sessions under the pretense of helping out. She might also try to show Percy how much she has changed or grown since they parted ways. This could involve her being more understanding of his needs and fears, something she perhaps wasn't so good at before.
Annabeth could show her determination by going out of her way to be in the same places as Percy. She'll be wherever he is, whether it's at his favorite spots or events he attends. And she'll always be there with a kind word or a helpful gesture, constantly reminding him of their connection. Also, she might face any danger or difficulty that comes her way while trying to get close to him again, showing that she's willing to do whatever it takes.
Well, she might face the obstacle of Percy's changed circumstances. He could be in a new place or situation that makes it difficult for her to reach him. Also, their past issues could be an obstacle. If they had a big fight before they split, those unresolved problems might make it hard for Annabeth to convince Percy to come back. And there could be social pressure in the fanfiction world, like friends or groups who think they're better off apart.
It could be because of a misunderstanding. In the fanfic, perhaps Annabeth was seen talking to an old acquaintance who Percy doesn't trust. He might jump to conclusions and get angry with her without really knowing the whole story. Like, if he thought she was plotting something behind his back with that person.
Annabeth could try to be more understanding of Percy's needs. In the fanfiction, she might give him space at first, but then slowly show up in his life in small, positive ways. Like bringing him his favorite food or helping him with a difficult task without being pushy.
One possibility is that Annabeth has to face some challenges to prove to Percy that she truly wants him back. She could be on a quest to find something important to Percy, like a lost item from his past. This would show her dedication. Also, there might be some new characters in the fanfic who try to stop her from getting back with Percy, adding more drama to the story.
There could also be external factors. For example, there might be a prophecy or some sort of danger that keeps Percy away from Annabeth. He might be so focused on dealing with this external threat that he doesn't have time or energy to think about getting back together with her.
The storyline could develop with Annabeth first reflecting on what went wrong in their relationship. She might write in a journal all the things she misses about Percy. Then she could take the bold step of showing up at Percy's new place unannounced. This would shock Percy but also open the door for communication.
A fanfic could focus on the wedding preparations. Annabeth is in charge of most of it, with Percy trying his best to help. There are comical moments like Percy accidentally setting the tablecloth on fire while trying to light the candles for the rehearsal dinner. And then on the wedding day, everything goes perfectly, with a beautiful sunset as they say 'I do'.