A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer.
User: A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet 4 inhabitants: Alexander, Grace, Avery, and Aurora. "Grace is not a knight" - Alexander. Grace expressed that Aurora is a knight and Grace is a knight. Avery was heard saying, "Grace is a knight if and only if Avery is a knight". "Avery is not a knave," Aurora claimed. So who is a knight and who is a knave? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags, for example <answer> (1) Zoey is a knight
(2) Oliver is a knight
(3)... </answer>.
Assistant: Let me solve this step by step.
<think> 
user
Assume Alexander is a knight.  No contradiction is found in their claim that Grace is not a knight.
 Grace cannot be a knight, because this would contradict the claim of Alexander that Grace is not a knight.
 Assume Grace is a knave.  No contradiction is found in their false claim that Aurora is a knight and Grace is a knight.
 Assume Aurora is a knight.  No contradiction is found in their claim that Avery is not a knave.
 Avery cannot be a knight, because this would contradict the claim of their own that Grace is a knight if and only if Avery is a knight.
 Avery cannot be a knave, because this would contradict the false claim of their own that Grace is a knight if and only if Avery is a knight.
 We have exhausted all possibilities for Avery, so let us go back and reconsider Aurora.
 Assume Aurora is a knave.  No contradiction is found in their false claim that Avery is not a knave.
 Avery cannot be a knight, because this would contradict the claim of their own that Grace is a knight if and only if Avery is a knight.
 Avery cannot be a knave, because this would contradict the false claim of their own that Grace is a knight if and only if Avery is a knight.
 We have exhausted all possibilities for Avery, Aurora, and Grace, so let us go back and reconsider Alexander.
 Assume Alexander is a knave.  No contradiction is found in their false claim that Grace is not a knight.
 Assume Grace is a knight.  No contradiction is found in their claim that Aurora is a knight and Grace is a knight.
 Assume Aurora is a knight.  No contradiction is found in their claim that Avery is not a knave.
 Assume Avery is a knight.  No contradiction is found in their claim that Grace is a knight if and only if Avery is a knight.
 Therefore, (1) Alexander is a knave
 (2) Grace is a knight
 (3) Avery is a knight
 (4) Aurora is a knight
A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer.
User: A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet 4 inhabitants: Alexander, Grace, Avery, and Aurora. "Grace is not a knight" - Alexander. Grace expressed that Aurora is a knight and Grace is a knight. Avery was heard saying, "Grace is a knight if and only if Avery is a knight". "Avery is not a knave," Aurora claimed. So who is a knight and who is a knave? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags, for example <answer> (1) Zoey is a knight
(2) Oliver is a knight
(3)... </answer>.
Assistant: Let me solve this step by step.
<think> 
Assume Alexander is a knight. No contradiction is found in their claim that Grace is not a knight.
Grace cannot be a knight, because this would contradict the claim of Alexander that Grace is not a knight.
Assume Grace is a knave. No contradiction is found in their false claim that Aurora is a knight and Grace is a knight.
Assume Aurora is a knight. No contradiction is found in their claim that Avery is not a knave.
Avery cannot be a knight, because this would contradict the claim of their own that Grace is a knight if and only if Avery is a knight.
Avery cannot be a knave, because this would contradict the false claim of their own that Grace is a knight if and only if Avery is a knight.
We have exhausted all possibilities for Avery, so let us go back and reconsider Aurora.
Assume Aurora is a knave. No contradiction is found in their false claim that Avery is not a knave.
Avery cannot be a knight, because this would contradict the claim of their own that Grace is a knight if and only if Avery is a knight.
Avery cannot be a knave, because this would contradict the false claim of their own that Grace is a knight if and only if Avery is a knight.
We have exhausted all possibilities for Avery, Aurora, and Grace, so let us go back and reconsider Alexander.
Assume Alexander is a knave. No contradiction is found in their false claim that Grace is not a knight.
Assume Grace is a knight. No contradiction is found in their claim that Aurora is a knight and Grace is a knight.
Assume Aurora is a knight. No contradiction is found in their claim that Avery is not a knave.
Assume Avery is a knight. No contradiction is found in their claim that Grace is a knight if and only if Avery is a knight.
Therefore, (1) Alexander is a knave
(2) Grace is a knight
(3) Avery is a knight
(4) Aurora is a knight </think>
<answer> (1) Alexander is a knave
(2) Grace is a knight
(3) Avery is a knight
(4) Aurora is a knight </answer>
Alexander Grace Avery Aurora
(1) Alexander is a knave
(2) Grace is a knight
(3) Avery is a knight
(4) Aurora is a knight
(('not', ('telling-truth', 1)), ('and', ('telling-truth', 3), ('telling-truth', 1)), ('<=>', ('telling-truth', 1), ('telling-truth', 2)), ('not', ('lying', 2)))
rule