Trace Duo Chat process flow with Langsmith

See merge request !146032



Merged-by: Jessie Young <jessieyoung@gitlab.com>
Approved-by: Mark Chao <mchao@gitlab.com>
Approved-by: Jessie Young <jessieyoung@gitlab.com>
Reviewed-by: Shinya Maeda <shinya@gitlab.com>
Reviewed-by: Mark Chao <mchao@gitlab.com>
Co-authored-by: Shinya Maeda <shinya@gitlab.com>