Update CallKit doc (#854)

Author: hiroshihorie

README.md

@@ -176,29 +176,39 @@ For more audio related information see the [Audio guide](./Docs/audio.md).
 
 ### Integration with CallKit
 
-To integrate with CallKit for background-triggered incoming calls, LiveKit's audio session must be synchronized with CallKit's audio session:
+When integrating with CallKit, proper timing and coordination between `AVAudioSession` and the SDK’s audio engine is crucial.
 
-1. Add `import LiveKitWebRTC` to your CallProvider file.
-2. In your `CXProviderDelegate` implementation, add the following:
+1. Disable the SDK’s automatic `AVAudioSession` configuration. also prevent the audio engine from starting outside CallKit’s `didActivate` and `didDeactivate` window.
 
 ```swift
-func provider(_ provider: CXProvider, didActivate audioSession: AVAudioSession){
-    LKRTCAudioSession.sharedInstance().audioSessionDidActivate(audioSession)
-    // ...
-}
-func provider(_ provider: CXProvider, didDeactivate audioSession: AVAudioSession) {
-    LKRTCAudioSession.sharedInstance().audioSessionDidDeactivate(audioSession)
-    // ...
-}
+// As early as possible, before connecting to a Room.
+AudioManager.shared.audioSession.isAutomaticConfigurationEnabled = false
+try AudioManager.shared.setEngineAvailability(.none)
 ```
 
-You would also want to turn off automatic `AVAudioSession` configuration.
+2. Coordinate audio engine availability with CallKit in your `CXProviderDelegate` implementation:
 
 ```swift
-AudioManager.shared.audioSession.isAutomaticConfigurationEnabled = false
+func provider(_: CXProvider, didActivate session: AVAudioSession) {
+  do {
+    try session.setCategory(.playAndRecord, mode: .voiceChat, options: [.mixWithOthers])
+    try AudioManager.shared.setEngineAvailability(.default)
+  } catch {
+    // Error
+  }
+}
+
+func provider(_: CXProvider, didDeactivate _: AVAudioSession) {
+  do {
+    try AudioManager.shared.setEngineAvailability(.none)
+  } catch {
+    // Error
+  }
+}
 ```
 
-For more audio related information see the [Audio guide](./Docs/audio.md).
+* See our [CallKit example](https://github.com/livekit-examples/swift-example-collection/tree/main/callkit) for full details.
+* For additional audio-related information, see the [Audio guide](./Docs/audio.md).
 
 ### iOS Simulator limitations