Alice was at the park when she saw an old woman struggling to carry a heavy basket. Alice quickly approached her and asked, "Would you like some help with that, ma'am?" The old woman smiled kindly and replied, "Thank you, dear, but I can manage. Although, if you could walk with me, that would be lovely." Alice agreed and walked beside the old woman, chatting along the way. As they reached the old woman’s house, she offered Alice some freshly baked cookies, saying, "Please, have some cookies as a thank-you for your kindness." Alice politely declined, "No, thank you, they look delicious, but I just ate." The old woman nodded and replied, "Maybe next time, dear."
What did Alice offer to the old woman at the beginning of the story?