JQ - 配列に要素を追加し、json全体を返します。

次のように配列に要素を追加しようとしています。

jq --arg root "arn:aws:iam::$ACCOUNT_ID:root" \
              '.Statement[] | select(.Sid=="Allow use of the key") \
              | .Principal.AWS |= . + [$root]' \
 <<<$policy

これが動作している間select。

コンテンツ$policy:

{
  "Version": "2012-10-17",
  "Id": "key-consolepolicy-3",
  "Statement": [
    {
      "Sid": "Enable IAM User Permissions",
      "Effect": "Allow",
      "Principal": {
        "AWS": "arn:aws:iam::*:root"
      },
      "Action": "kms:*",
      "Resource": "*"
    },
    {
      "Sid": "Allow access for Key Administrators",
      "Effect": "Allow",
      "Principal": {
        "AWS": "arn:aws:iam::*:user/*"
      },
      "Action": [
        "kms:Create*",
        "kms:Describe*",
        "kms:Enable*",
        "kms:List*",
        "kms:Put*",
        "kms:Update*",
        "kms:Revoke*",
        "kms:Disable*",
        "kms:Get*",
        "kms:Delete*",
        "kms:TagResource",
        "kms:UntagResource"
      ],
      "Resource": "*"
    },
    {
      "Sid": "Allow use of the key",
      "Effect": "Allow",
      "Principal": {
        "AWS": [
          "arn:aws:iam::*:root"
        ]
      },
      "Action": [
        "kms:Encrypt",
        "kms:Decrypt",
        "kms:DescribeKey",
        "kms:ReEncrypt*",
        "kms:GenerateDataKey*",
        "kms:CreateGrant"
      ],
      "Resource": "*"
    }
  ]
}

予想される結果：

{
  "Version": "2012-10-17",
  "Id": "key-consolepolicy-3",
  "Statement": [
    {
      "Sid": "Enable IAM User Permissions",
      "Effect": "Allow",
      "Principal": {
        "AWS": "arn:aws:iam::*:root"
      },
      "Action": "kms:*",
      "Resource": "*"
    },
    {
      "Sid": "Allow access for Key Administrators",
      "Effect": "Allow",
      "Principal": {
        "AWS": "arn:aws:iam::*:user/*"
      },
      "Action": [
        "kms:Create*",
        "kms:Describe*",
        "kms:Enable*",
        "kms:List*",
        "kms:Put*",
        "kms:Update*",
        "kms:Revoke*",
        "kms:Disable*",
        "kms:Get*",
        "kms:Delete*",
        "kms:TagResource",
        "kms:UntagResource"
      ],
      "Resource": "*"
    },
    {
      "Sid": "Allow use of the key",
      "Effect": "Allow",
      "Principal": {
        "AWS": [
          "arn:aws:iam::*:root",
          "arn:aws:iam::$root:root" <-------------
        ]
      },
      "Action": [
        "kms:Encrypt",
        "kms:Decrypt",
        "kms:DescribeKey",
        "kms:ReEncrypt*",
        "kms:GenerateDataKey*",
        "kms:CreateGrant"
      ],
      "Resource": "*"
    }
  ]
}

追加された要素の初期jsonをどのように返しますか？