docs(background-task): enhance background_output tool description with full_session parameter

src/tools/background-task/constants.ts

@@ -2,6 +2,6 @@ export const BACKGROUND_TASK_DESCRIPTION = `Run agent task in background. Return
 
 Use \`background_output\` to get results. Prompts MUST be in English.`
 
-export const BACKGROUND_OUTPUT_DESCRIPTION = `Get output from background task. System notifies on completion, so block=true rarely needed.`
+export const BACKGROUND_OUTPUT_DESCRIPTION = `Get output from background task. Use full_session=true to fetch session messages with filters. System notifies on completion, so block=true rarely needed.`
 
 export const BACKGROUND_CANCEL_DESCRIPTION = `Cancel running background task(s). Use all=true to cancel ALL before final answer.`

src/tools/background-task/index.ts

@@ -5,3 +5,4 @@ export {
 
 export type * from "./types"
 export * from "./constants"
+export type { BackgroundOutputClient, BackgroundOutputManager, BackgroundCancelClient } from "./tools"