Group documents from another collection into the field of array of nested document

Working with Data
,
1

I have 2 collections, messages and chats. How can I use aggregation to group and messages of the same conversation id into an array and put it in the field of array of conversations? In addition, I want to sort the messeages by its timestamp and remove the conversationId.

messages schema:

conversationId: String,
content: String,
timestamp: Date

chats schema:

userId: ObjectId,
conversations: [{
  id: String 
  name: String
}]

Example output:

{
userId: ObjectId,
conversations: [
  {
    id: "conv1",
    name: "a",
    messages: [{content: "bruh", timestamp: Date}, ...]
  },
  {
    id: "conv2",
    name: "b",
    messages: [{content: "happy happy happy", timestamp: Date}, ...]
  }
]
}

I tried this but I do not know what to do next. Perhaps $bucket?

db.chats.aggregate([
  {
    $lookup: {
      "from": "messages",
      "localField": "conversations.id",
      "foreignField": "conversationId",
      "pipeline": [
        {
          $project: {
            "conversationId": 0
          }
        },
        {
          $sort: {
            "timestamp": 1,
            "_id": 1
          }
        }
      ],
      "as": "messages"
    }
  }
])