fix(web): persist thinking, title sessions distinctly, and send follow-ups from the answer

Thinking disappeared on failure and never reached session storage. Keep the
sanitized chain on disk and on errors, and regroup the sidebar around reports,
charts, favorites, and dated history titles.

Co-authored-by: Cursor <cursoragent@cursor.com>

frontend/tests/agent-activity-progress.test.ts

@@ -3,6 +3,7 @@ import test from "node:test";
 
 import { agentGenerationSettings, AGENT_MAX_OUTPUT_TOKENS } from "../src/lib/agent-generation-settings.ts";
 import {
+  activityCompletedSteps,
   activityCompletedTrail,
   activityElapsedLabel,
   nextActivityView,
@@ -77,6 +78,10 @@ test("completed-step trail stays short and drops while composing", () => {
     label: "正在组织回答…",
   }, 20);
   assert.equal(composing.completedTrail, undefined);
+  assert.deepEqual(
+    activityCompletedSteps("已完成：读取分析方法 · 计算本命盘"),
+    ["读取分析方法", "计算本命盘"],
+  );
 });
 
 test("live rectification labels name the actual public tool", () => {