feat(background-task): show original prompt and last message in running task status

- Add prompt field to BackgroundTask to store original prompt
- Add lastMessage/lastMessageAt to TaskProgress for real-time monitoring
- Extract last assistant message during polling
- Update formatTaskStatus() to display prompt (truncated 300 chars) and
  last message (truncated 500 chars) with timestamp

🤖 GENERATED WITH ASSISTANCE OF [OhMyOpenCode](https://github.com/code-yeongyu/oh-my-opencode)

src/features/background-agent/manager.ts

@@ -59,6 +59,7 @@ export class BackgroundManager {
       parentSessionID: input.parentSessionID,
       parentMessageID: input.parentMessageID,
       description: input.description,
+      prompt: input.prompt,
       agent: input.agent,
       status: "running",
       startedAt: new Date(),
@@ -316,7 +317,7 @@ export class BackgroundManager {
         if (!messagesResult.error && messagesResult.data) {
           const messages = messagesResult.data as Array<{
             info?: { role?: string }
-            parts?: Array<{ type?: string; tool?: string; name?: string }>
+            parts?: Array<{ type?: string; tool?: string; name?: string; text?: string }>
           }>
           const assistantMsgs = messages.filter(
             (m) => m.info?.role === "assistant"
@@ -324,6 +325,7 @@ export class BackgroundManager {
 
           let toolCalls = 0
           let lastTool: string | undefined
+          let lastMessage: string | undefined
 
           for (const msg of assistantMsgs) {
             const parts = msg.parts ?? []
@@ -332,6 +334,9 @@ export class BackgroundManager {
                 toolCalls++
                 lastTool = part.tool || part.name || "unknown"
               }
+              if (part.type === "text" && part.text) {
+                lastMessage = part.text
+              }
             }
           }
 
@@ -341,6 +346,10 @@ export class BackgroundManager {
           task.progress.toolCalls = toolCalls
           task.progress.lastTool = lastTool
           task.progress.lastUpdate = new Date()
+          if (lastMessage) {
+            task.progress.lastMessage = lastMessage
+            task.progress.lastMessageAt = new Date()
+          }
         }
       } catch (error) {
         log("[background-agent] Poll error for task:", { taskId: task.id, error })

src/features/background-agent/types.ts

@@ -8,6 +8,8 @@ export interface TaskProgress {
   toolCalls: number
   lastTool?: string
   lastUpdate: Date
+  lastMessage?: string
+  lastMessageAt?: Date
 }
 
 export interface BackgroundTask {
@@ -16,6 +18,7 @@ export interface BackgroundTask {
   parentSessionID: string
   parentMessageID: string
   description: string
+  prompt: string
   agent: string
   status: BackgroundTaskStatus
   startedAt: Date