Task/Issue URL:
https://app.asana.com/1/137249556945/project/1202552961248957/task/1215449563262684?focus=true

### Description

The post-idle session wide event (`wide_post_idle_session`) now records
two user actions it previously missed: selecting a Duck.ai chat
suggestion (new `chat_selected` outcome) and opening the tab switcher
from the NTP-after-idle hatch (`tab_switcher_selected`). Typed Duck.ai
prompts are also now recorded as `bar_used`, like search. This closes
instrumentation gaps so the NTP-vs-Last-Used-Tab after-idle comparison
stays accurate and matches the equivalent iOS measurement.

### Steps to test this PR

_NTP-after-idle_
- [ ] Background the app past the idle threshold, reopen → confirm a
session starts with `surface=ntp`
- [ ] Type a query and submit a search → confirm the session finishes
with `reason=bar_used`
- [ ] Trigger a fresh NTP-after-idle, toggle to Duck.ai, type a prompt
and submit → confirm `reason=bar_used`
- [ ] Trigger a fresh NTP-after-idle, tap a Duck.ai chat suggestion →
confirm `reason=chat_selected`
- [ ] Trigger a fresh NTP-after-idle, open the tab switcher from the
hatch's tab button → confirm `reason=tab_switcher_selected`

_LUT-after-idle_
- [ ] Background past the idle threshold, reopen onto the last tab →
confirm a session starts with `surface=lut`
- [ ] Submit a search or Duck.ai prompt → confirm `reason=bar_used`
- [ ] Trigger a fresh LUT-after-idle, tap a Duck.ai chat suggestion →
confirm `reason=chat_selected`

### UI changes

n/a