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Json<start>= Object | Array;
Object= "{" [ Field { "," Field } ] "}";
Field= string ":" Value;
Array= "[" [ Value { "," Value } ] "]";
Value<collapsed>= 
    string    |
    number    |
    Object    |
    Array     |
    Boolean   |
    null      ;
Boolean= true|false;
number= '\-?(0|[1-9][0-9]*)(\.[0-9]+)?([Ee][\+\-]?[0-9]+)?';
string= '"([^\n"\\]|\\([btrnf"\\/]|(u[0-9A-Fa-f]{4})))*"';
true= "true";
false= "false";
null= "null";
lbracket<collapsed>= "[";
rbracket<collapsed>= "]";
lbrace<collapsed>= "{";
rbrace<collapsed>= "}";
colon<collapsed>= ":";
comma<collapsed>= ",";
whitespace<hidden>= '[\n\r\t ]+';

public class StringDictionaryConverter : JsonConverter
{
    public override bool CanRead => true;
    public override bool CanWrite => true;
    public override bool CanConvert(Type objectType) => typeof(StringDictionary).IsAssignableFrom(objectType);
    
    public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
    {
        // Serialize the StringDictionary "properly":
        var dictionary = (StringDictionary)value;
        writer.WriteStartObject();
        foreach (DictionaryEntry entry in dictionary)
        {
            writer.WritePropertyName((string)entry.Key);
            writer.WriteValue(entry.Value);
        }
        writer.WriteEndObject();
    }
    
    public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
    {
        if (reader.TokenType == JsonToken.Null) return null;
        
        var dictionary = existingValue as StringDictionary ?? (StringDictionary)serializer.ContractResolver.ResolveContract(objectType).DefaultCreator();
        switch (reader.TokenType)
        {
            case JsonToken.StartObject:
            {
                // Serialized with converter:
                var values = serializer.Deserialize<Dictionary<string, string>>(reader);
                foreach (var pair in values)
                {
                    dictionary.Add(pair.Key, pair.Value);
                }
                break;
            }
            case JsonToken.StartArray:
            {
                // Serialized without converter:
                var values = serializer.Deserialize<List<KeyValuePair<string, string>>>(reader);
                foreach (var pair in values)
                {
                    dictionary.Add(pair.Key, pair.Value);
                }
                break;
            }
            default:
            {
                throw new JsonSerializationException($"Unknown token {reader.TokenType} at path {reader.Path}");
            }
        }
        
        return dictionary;
    }
}

var source = new StringDictionary
{
    ["xml"] = "sucks",
    ["json"] = "rules",
};

string json = JsonConvert.SerializeObject(source, new StringDictionaryConverter());
var result = JsonConvert.DeserializeObject<StringDictionary>(json, new StringDictionaryConverter());