[Feature Request] Add option to log thinking traces to file in --print mode

Feature Description
I'm wanting to log thinking traces to a file when in --print mode (as that currently only prints out the last message) so I can later analyze what went wrong with a particular instruction, but there doesn't seem to be a way of doing so atm.

Any chance you could implement this? (Claude Code itself can apparently implement a hacky version of this on minified javascript...)

Environment Info

• Platform: linux
• Terminal: gnome-terminal
• Version: 2.0.55
• Feedback ID: cff3e46e-0abb-481b-afa3-2993ee3ee1af