C# - Types of asynchronous background code execution in REST API endpoint?

I am implementing a REST API endpoint in ASP.NET, that should do the following:

1. Asynchronously start a long-running algorithm
2. Return 202 Accepted immediately after starting the algorithm (no await)

This is what the endpoint looks like:

app.MapPut(
    "/api/instance/{id}",
    (int id, InstanceDto instanceDto, IInstanceRunningService service) =>
    {
        // Start the execution asynchronously and return ASAP
        service.RunInstance(instanceDto);
        return TypedResults.Accepted($"/api/instance/{id}");
    })

This is what the RunInstance method in the service looks like:

public async Task RunInstance(InstanceDto instanceDto)
{
    // Asynchronously call another API that we are starting the execution of the algorithm
    // We do not need to wait for the response, we just want to start the algorithm ASAP (again no await)
    _ = this.AnotherApiClient.PutAsync(instanceDto);

    // Compute the algorithm and await the result
    var result = await Algorithm.Run(instanceDto.Parameters);
    
    // Finally call the other API again to notify that the computation finished
    await this.AnotherApiClient.PutAsync(instanceDto);
}

The order of operations I want to achieve is:

1. Return 202 Accepted
2. Notify another API that computation is starting (we don't care how this API call ends, because we will be sending another one in step 4.)
3. Compute the algorithm
4. Notify another API that computation has finished

I can't figure out the best way to achieve this. There are so many options:

• service.RunInstance(instanceDto);
• service.RunInstance(instanceDto).Start();
• Task.Run(() => service.RunInstance(instanceDto))
• Make RunInstance an asynchronous delegate and do delegate.BeginInvoke()
• Create a Thread or use ThreadPool
• Maybe implement the service differently with more async methods