Count number of elements in mongo array with filter for specific documents [duplicate]

I have a simple collection with documents of vendors that looks like this:

{
  "name": "Nick's Burger",
  "country": "AT",
  "branches": [
    {
      "branchId": "12535",
      "branchName": "Nick's Burger Branch One",
      "branchAddress": "Somewhere in austria"
    },
    {
      "branchId": "38267",
      "branchName": "Nick's Burger Branch Two",
      "branchAddress": "Somewhere else in austria"
    }
  ]
}

For each of the vendors there is a list of "branches", and I want to count how many branches all the vendors from "country" AT have.

tried something like:

db.vendors.find({country: "AT"}, {$size: {branches: 1}}).count()

but that only gives me the count of vendors from AT...