feat: Update and submit user value

While this feature is directly related towards shiny.ui.Chat, I'm posting the issue here as it is targeted towards chatlas.

Goal: Set and submit user value and display streaming result as if the user had done it.

Using await chat.chat_async("NEW_MSG") does not update the Shiny UI, but it'd be really cool if it did.

Current work around:

@reactive.effect
@reactive.event(trigger)
async def _():
    chat_ui.update_user_input(value="who is the tallest skywalker")

    add_chat_entry = ui.HTML("""
    <script>
        setTimeout(async () => {
            const enterEvent = new KeyboardEvent('keydown', {
                key: 'Enter',
                code: 'Enter',
                keyCode: 13,
                which: 13,
            });

            // Dispatch the 'Enter' event on the input element
            console.log("Dispatching Enter event");
            document.querySelector("textarea#chat_user_input").dispatchEvent(enterEvent);
        }, 500);
    </script>
            """)
    ui.insert_ui(add_chat_entry, "body")

Start with the chat.chat_async("NEW_MSG") path and the Shiny UI integration around chat_ui.update_user_input and textarea#chat_user_input. Compare the current workaround with the desired behavior, then verify that programmatic user values update the UI and display the streaming response as if submitted interactively.