This PR adds a model for supporting background responses (long-running operations), updates the chat completion model to use it, and integrates this functionality intoOpenAIResponsesChatClient.

Background responses use a continuation token returned by theGetResponseAsync andGetStreamingResponseAsync methods in theChatResponse andChatResponseUpdate classes, respectively, when background responses are enabled and supported by the chat client.

The continuation token contains all necessary details to enable polling for a background response using the non-streamingGetResponseAsync method. It also allows resuming a streamed background response with theGetStreamingResponseAsync method if the stream is interrupted. In both cases, the continuation token obtained from the initial chat result or the streamed update received before the interruption should be supplied as input for follow-up calls.

When a background response has completed, failed, or cannot proceed further (for example, when user input is required), the continuation token returned by either method will be null, signaling to consumers that processing is complete and there is nothing to poll or resume. The result returned by either method can then be used for further processing.

Non-streaming API:

varchatOptions=newChatOptions{BackgroundResponsesOptions=new(){Allow=true}};// Get initial response with continuation tokenvarresponse=awaitchatClient.GetResponseAsync("What's the biggest animal?",chatOptions);// Continue to poll until the final response is receivedwhile(response.ContinuationTokenis notnull){chatOptions.ContinuationToken=response.ContinuationToken;response=awaitchatClient.GetResponseAsync([],chatOptions);}Console.WriteLine(response);

Streaming API:

ChatOptionschatOptions=new(){BackgroundResponsesOptions=newBackgroundResponsesOptions{Allow=true},};stringresponse="";ResumptionToken?continuationToken=null;awaitforeach(varupdateinchatClient.GetStreamingResponseAsync("What is the capital of France?",chatOptions)){response+=update;// Simulate an interruptioncontinuationToken=update.ContinuationToken;break;}// Resume streaming from the point captured by the continuation tokenchatOptions.ContinuationToken=continuationToken!awaitforeach(varupdateinchatClient.GetStreamingResponseAsync([],chatOptions)){response+=update;}Console.WriteLine(response);

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CC:@westey-m