C# How to Serialize/Deserialize an IEnumerator from a Yield function (to Json?)

I'm trying to serialize/deserialize an IEnumerator generated from a function using yield. I would like to serialize the IEnumerator at any iteration, I don't want to force it to generate all of it values.

I know that the yield keyword generate a class behind the scene, and that why I'm using it, to avoid manually writing iterator, and also to make the code cleaner.

My goal is to make a small game engine similar to Nick Gravelyn - The magic of yield, where each GameElement generate an iterator about his behavior, allowing the programmer to easily control timings (because the iterator allow to interrupt and continue a script). I want to try to add a multiplayer layer on top of that, that why I need to serialize/deserialize an IEnumerator.

Forcing the IEnumerator to generate all of his values also force the game to update, this should be avoided.

My first tentative was somethings like :

using System;
using System.Collections.Generic;
using System.Text.Json;

namespace SerializeTest
{
    class Program
    {
        public static IEnumerator<int> CountTo(int end)
        {
            for(int i = 1; i <= end; i++) 
            {
                Console.WriteLine("i = " + i);
                yield return i;
            }
        }

        static void Main(string[] args)
        {
            IEnumerator<int> e = CountTo(5);
            string json = JsonSerializer.Serialize(e);
            Console.WriteLine(json);
            var f = JsonSerializer.Deserialize<IEnumerator<int>> (json); // crash because interface

            while (f.MoveNext());
        }
    }
}

But deserializing an interface (IEnumerator<int>) is illegal, we need to know the Type of the object behind IEnumerator<int>.

Since the type is generated by the compiler, I try to use reflection over the Deserialize method in order to call it with the correct type generated by the function CountTo() (SerializeTest.Program+<CountTo>d__0 if you are curious)

So my second tentative was somethings like :

using System;
using System.Collections.Generic;
using System.Reflection;
using System.Text.Json;

namespace SerializeTest
{
    class Program
    {
        public static IEnumerator<int> CountTo(int end)
        {
            for(int i = 1; i <= end; i++) 
            {
                Console.WriteLine("i = " + i);
                yield return i;
            }
        }

        public static T Deserialize<T>(string json) => JsonSerializer.Deserialize<T>(json); // still crash

        static void Main(string[] args)
        {
            IEnumerator<int> e = CountTo(5);

            string json = JsonSerializer.Serialize(e);
            Console.WriteLine(json);

            Type eType = e.GetType();
            MethodInfo method = typeof(Program).GetMethod("Deserialize");
            MethodInfo genericMethod = method.MakeGenericMethod(eType);

            object fObj = genericMethod.Invoke(null, new object[] { json });
            var f = (IEnumerator<int>)fObj;
            while (f.MoveNext()) ;
        }
    }
}

but it still crash when calling JsonSerializer.Deserialize() even if the type is now correct and not an interface.

(System.InvalidOperationException : 'Each parameter in constructor 'Void .ctor(Int32)' on type 'SerializeTest.Program+<CountTo>d__0' must bind to an object property or field on deserialization. Each parameter name must match with a property or field on the object. The match can be case-insensitive.' )

The format don't have any importance (json, xml...) as long as it is possible to send it to another computer and deserialize it back.

I naively hope that there is a way to Serialize/Deserialize an IEnumerator because a lot of cool stuff can be done with it, although I have some serious doubts about such a things to be possible.

Thank for reading so far.