Output a specific result from array of objects aggregation

I have a data looking like this:

{
  "_id": ObjectId("620ca4e292a990273446cf7b"),
  "data": [
    {
      "flagging": "BUSINESS",
      "information": [
        {
          "room": 1,
          "status": "AVAILABLE"
        },
        {
          "room": 2,
          "status": "BUSY"
        }
      ]
    },
    {
      "flagging": "PRIVATE",
      "information": [
        {
          "room": 1,
          "status": "AVAILABLE"
        },
        {
          "room": 2,
          "status": "CLEANING"
        }
      ]
    },
    {
      "flagging": "PUBLIC",
      "information": [
        {
          "room": 1,
          "status": "SERVICE"
        },
        {
          "room": 2,
          "status": "AVAILABLE"
        }
      ]
    }
  ],
  "createdAt": "2022-02-16T07:16:50.674Z"
}

I'm trying to make a service whether user can input a certain flagging, and the data with the corresponding flagging will show as a result. I tried using this code below but what I get are all the datas, not the filtered ones.

db.coll.aggregate([
    {
        $match: { _id: ObjectId("620ca4e292a990273446cf7b") },
    },
    {
        $project: { _id: 0, "data": 1 }
    }
])

What should I change in the code so I can get the following result (if the requested flagging are BUSINESS and PUBLIC)?

{
  "data": [
    {
      "flagging": "BUSINESS",
      "information": [
        {
          "room": 1,
          "status": "AVAILABLE"
        },
        {
          "room": 2,
          "status": "BUSY"
        }
      ]
    },
    {
      "flagging": "PUBLIC",
      "information": [
        {
          "room": 1,
          "status": "SERVICE"
        },
        {
          "room": 2,
          "status": "AVAILABLE"
        }
      ]
    }
  ]
}

Thanks for the help