feat(ui): track solve_run analytics event - #20

Bugbot Analysis Progress (4m 55s elapsed)

✅ Gathered PR context (3s)
✅ Completed bug detection (4m 52s)
✅ Posted analysis results (1s)

Final Result: Bugbot completed review - no issues found! ✅

Request ID: c5d18ee4-15b4-4b65-9d36-e63e24a04a13