Show live background task counts

app/androidApp/src/main/kotlin/com/example/aiapp/Api.kt

@@ -225,6 +225,8 @@ data class SessionSummary(
     val usageProvider: String?,
     val status: String,
     val lastActivity: Double,
+    /** Latest measured number of live background tasks; zero also covers older servers. */
+    val backgroundTasks: Int,
     /**
      * How many subagents this session has, however their own status now reads.
      *
@@ -263,6 +265,7 @@ private fun parseSession(session: JSONObject) =
         usageProvider = session.optString("usageProvider").ifEmpty { null },
         status = session.getString("status"),
         lastActivity = session.getDouble("lastActivity"),
+        backgroundTasks = session.optInt("backgroundTasks", 0),
         subagents = session.optInt("subagents", 0),
     )