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>Hint: Please answer the question and provide the correct option letter, e.g., A, B, C, D, E, at the end.
Question: Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<image1>
Choices:
(A) $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(B) $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right) \neq \operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(C) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)>\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(D) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)<\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(E) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$ 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
Hint: Please answer the question and provide the correct option letter, e.g., A, B, C, D, E, at the end.
Question: Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<image1>
Choices:
(A) $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(B) $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right) \neq \operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(C) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)>\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(D) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)<\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
(E) $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$ 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