Move the browsing context out of the system prompt and injecting it
directly into the last user message instead.
Previously (before this PR), browsing context change, update system
prompt then break whole conversation history ... and caching. Now,
browsing context is sent with last message only if changed.
"Benchmark" this PR vs main :
- same conv with 3-4 turns - 60% -> 85% cache ratio || 0.31 credits ->
0.13
