Split array-like string values in MongoDB aggregation

A field in my data is stored as a string instead of a real nested array:

{
  "uuid": "yxcvb", 
  "product": "[{\"pid\":\"4f76c06825aa486db9a7a5cedbc2ea19\",\"size\":\"S\"},{\"pid\":\"d3055e0a1bb040dba65674b9733bf4ce\",\"size\":\"M\"}]",
  "timestamp":{"$date":"2023-01-01T11:25:47+00:00"}
},
{
  "uuid": "asdfg", 
  "product": "[{\"pid\":\"369b33cc768d496193b18d657d706920\",\"size\":\"M\"}]",
  "timestamp":{"$date":"2023-01-02T11:25:47+00:00"}
},
{
  "uuid": "qwert", 
  "product": "[{\"pid\":\"13325cb71fd7413c8f3e18caf20b5d6d\",\"size\":\"S\"},{\"pid\":\"ac431397d49d449cab44d98dc13ec57c\",\"size\":\"M\"},{\"pid\":\"ba35b9f804a44e15bf197d3ef671dc34\",\"size\":\"L\"}]",
  "timestamp":{"$date":"2023-01-03T11:25:47+00:00"}
}

/// "product" has a maximum of 4 dict-like pid-size units

I would like to know the count of each size and group by timestamp. Expected output:

[
  {
    "week": 52,
    "sizeS": 1
  },
  {
    "week": 52,
    "sizeM": 1
  },
  {
    "week": 1,
    "sizeS": 1
  },
  {
    "week": 1,
    "sizeM": 2
  },
  {
    "week": 1,
    "sizeL": 1
  }
]

To realize this, the following problems need to be solved:

1. Turn the array-like string into a real string, then I could use $unwind.
2. If 1 cannot be achieved, extract all pid and size values in a document, and save only the value after the colon (e.g. "4f76c06825aa486db9a7a5cedbc2ea19", "M").
3. After 2, count total records of each size within the grouped timeframe (week).

You can use $function to parse the json strings and use it as normal mongodb fields.

db.collection.aggregate([
  {
    "$set": {
      "product": {
        "$function": {
          "body": "function(product) {return JSON.parse(product)}",
          "args": [
            "$product"
          ],
          "lang": "js"
        }
      }
    }
  },
  {
    "$unwind": "$product"
  },
  {
    $group: {
      _id: {
        week: {
          "$isoWeek": "$timestamp"
        },
        size: "$product.size"
      },
      count: {
        $sum: 1
      }
    }
  },
  {
    $sort: {
      "_id.week": -1
    }
  }
])

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