feat: expose worktree path and branch in task API and UI

- Add WorktreePath and WorktreeBranch fields to TaskDTO
- Add WorktreeLookup interface to TaskController for enriching task responses
- Add GetWorktreeInfo method to worktree.Manager implementing WorktreeLookup
- Wire worktree manager to task controller in main.go
- Add worktree_path and worktree_branch to frontend Task type
- Display worktree path with copy button in TaskTopBar when worktree exists
- Show worktree branch in tooltip for easy reference

apps/backend/internal/task/controller/task.go

@@ -7,14 +7,36 @@ import (
 	"github.com/kandev/kandev/internal/task/service"
 )
 
+// WorktreeLookup provides worktree information for tasks.
+type WorktreeLookup interface {
+	// GetWorktreeInfo returns worktree path and branch for a task, or nil if no worktree exists.
+	GetWorktreeInfo(ctx context.Context, taskID string) (path, branch *string)
+}
+
 type TaskController struct {
-	service *service.Service
+	service         *service.Service
+	worktreeLookup  WorktreeLookup
 }
 
 func NewTaskController(svc *service.Service) *TaskController {
 	return &TaskController{service: svc}
 }
 
+// SetWorktreeLookup sets the worktree lookup for enriching task responses.
+func (c *TaskController) SetWorktreeLookup(lookup WorktreeLookup) {
+	c.worktreeLookup = lookup
+}
+
+// enrichWithWorktree adds worktree info to a TaskDTO if available.
+func (c *TaskController) enrichWithWorktree(ctx context.Context, task *dto.TaskDTO) {
+	if c.worktreeLookup == nil {
+		return
+	}
+	path, branch := c.worktreeLookup.GetWorktreeInfo(ctx, task.ID)
+	task.WorktreePath = path
+	task.WorktreeBranch = branch
+}
+
 func (c *TaskController) ListTasks(ctx context.Context, req dto.ListTasksRequest) (dto.ListTasksResponse, error) {
 	tasks, err := c.service.ListTasks(ctx, req.BoardID)
 	if err != nil {
@@ -25,7 +47,9 @@ func (c *TaskController) ListTasks(ctx context.Context, req dto.ListTasksRequest
 		Total: len(tasks),
 	}
 	for _, task := range tasks {
-		resp.Tasks = append(resp.Tasks, dto.FromTask(task))
+		taskDTO := dto.FromTask(task)
+		c.enrichWithWorktree(ctx, &taskDTO)
+		resp.Tasks = append(resp.Tasks, taskDTO)
 	}
 	return resp, nil
 }
@@ -35,7 +59,9 @@ func (c *TaskController) GetTask(ctx context.Context, req dto.GetTaskRequest) (d
 	if err != nil {
 		return dto.TaskDTO{}, err
 	}
-	return dto.FromTask(task), nil
+	taskDTO := dto.FromTask(task)
+	c.enrichWithWorktree(ctx, &taskDTO)
+	return taskDTO, nil
 }
 
 func (c *TaskController) CreateTask(ctx context.Context, req dto.CreateTaskRequest) (dto.TaskDTO, error) {