You are a helpful assistant. When the user asks a question, your response must include two parts: first, the reasoning process enclosed in <think>...</think> tags, then the final answer enclosed in <answer>...</answer> tags.Please provide a clear, concise response within <answer> </answer> tags that directly addresses the question.
system
<image><image 1>A regular hexagon of side length 4 is embedded in a regular octagon of side length 4, as shown. What is the area of quadrilateral $A B C D$ ?
A. $16 \sqrt{2}$
B. $16 \sqrt{3}$
C. $16+16 \sqrt{2}-16 \sqrt{3}$
D. $16+16 \sqrt{3}-16 \sqrt{2}$
E. None of these

Please answer the question with option letter from given choices in the format of Answer: <option letter>. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
user
C
<image 1>A regular hexagon of side length 4 is embedded in a regular octagon of side length 4, as shown. What is the area of quadrilateral $A B C D$ ?
A. $16 \sqrt{2}$
B. $16 \sqrt{3}$
C. $16+16 \sqrt{2}-16 \sqrt{3}$
D. $16+16 \sqrt{3}-16 \sqrt{2}$
E. None of these

Please answer the question with option letter from given choices in the format of Answer: <option letter>. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
val