feat: searchable timezone select & bot timezone priority

- Add reusable TimezoneSelect component with search and UTC offset labels
- Replace plain Select with searchable TimezoneSelect in profile settings,
  bot settings, and browser context settings
- Move bot timezone setting from header dialog into bot settings tab
- Resolve timezone with bot > user > system priority for all LLM-facing
  time formatting (user message header, system prompt, heartbeat, tools,
  memory extraction)
- Format tool output timestamps (history, contacts) in resolved timezone

internal/agent/tools/contacts.go

@@ -67,7 +67,7 @@ func (p *ContactsProvider) Tools(_ context.Context, session SessionContext) ([]s
 						"conversation_type": r.ConversationType,
 						"target":            r.ReplyTarget,
 						"conversation_id":   r.ConversationID,
-						"last_active":       r.UpdatedAt.Format("2006-01-02T15:04:05Z"),
+						"last_active":       sess.FormatTime(r.UpdatedAt),
 					}
 					if len(r.Metadata) > 0 {
 						if v, ok := r.Metadata["conversation_name"].(string); ok && v != "" {

internal/agent/tools/history.go

@@ -166,8 +166,8 @@ func (p *HistoryProvider) execListSessions(ctx context.Context, sess SessionCont
 			"platform":          s.ChannelType,
 			"route_id":          s.RouteID,
 			"conversation_type": s.RouteConversationType,
-			"last_active":       s.UpdatedAt.Format(time.RFC3339),
-			"created_at":        s.CreatedAt.Format(time.RFC3339),
+			"last_active":       sess.FormatTime(s.UpdatedAt),
+			"created_at":        sess.FormatTime(s.CreatedAt),
 		}
 
 		if m := s.RouteMetadata; len(m) > 0 {
@@ -261,7 +261,7 @@ func (p *HistoryProvider) execSearchMessages(ctx context.Context, sess SessionCo
 			"session_id": row.SessionID.String(),
 			"role":       row.Role,
 			"text":       text,
-			"created_at": row.CreatedAt.Time.Format(time.RFC3339),
+			"created_at": sess.FormatTime(row.CreatedAt.Time),
 		}
 		if dbpkg.TextToString(row.Platform) != "" {
 			entry["platform"] = dbpkg.TextToString(row.Platform)

internal/agent/tools/types.go

@@ -6,6 +6,7 @@ import (
 	"fmt"
 	"math"
 	"strings"
+	"time"
 
 	sdk "github.com/memohai/twilight-ai/sdk"
 )
@@ -28,6 +29,15 @@ type SessionContext struct {
 	SupportsImageInput bool
 	IsSubagent         bool
 	Skills             map[string]SkillDetail
+	TimezoneLocation   *time.Location
+}
+
+// FormatTime formats a time.Time using the session timezone (falls back to UTC).
+func (s SessionContext) FormatTime(t time.Time) string {
+	if s.TimezoneLocation != nil {
+		t = t.In(s.TimezoneLocation)
+	}
+	return t.Format(time.RFC3339)
 }
 
 // ToolProvider supplies a set of tools for the agent.