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: Aurora, Lily, Ava, and Daniel. Aurora expressed that Daniel is a knight and Ava is a knight. In a statement by Lily: "Ava is a knight if and only if Aurora is a knave". Ava commented, "If Daniel is a knave then Lily is a knave". Daniel commented, "Ava is a knight or Lily is a knave". 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 Aurora is a knight.  No contradiction is found in their claim that Daniel is a knight and Ava is a knight.
 Assume Daniel is a knight.  No contradiction is found in their claim that Ava is a knight or Lily is a knave.
 Assume Ava is a knight.  No contradiction is found in their claim that If Daniel is a knave then Lily is a knave.
 Lily cannot be a knight, because this would contradict the claim of their own that Ava is a knight if and only if Aurora is a knave.
 Assume Lily is a knave.  No contradiction is found in their false claim that Ava is a knight if and only if Aurora is a knave.
 Therefore, (1) Aurora is a knight
 (2) Lily is a knave
 (3) Ava is a knight
 (4) Daniel 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: Aurora, Lily, Ava, and Daniel. Aurora expressed that Daniel is a knight and Ava is a knight. In a statement by Lily: "Ava is a knight if and only if Aurora is a knave". Ava commented, "If Daniel is a knave then Lily is a knave". Daniel commented, "Ava is a knight or Lily is a knave". 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 Aurora is a knight. No contradiction is found in their claim that Daniel is a knight and Ava is a knight.
Assume Daniel is a knight. No contradiction is found in their claim that Ava is a knight or Lily is a knave.
Assume Ava is a knight. No contradiction is found in their claim that If Daniel is a knave then Lily is a knave.
Lily cannot be a knight, because this would contradict the claim of their own that Ava is a knight if and only if Aurora is a knave.
Assume Lily is a knave. No contradiction is found in their false claim that Ava is a knight if and only if Aurora is a knave.
Therefore, (1) Aurora is a knight
(2) Lily is a knave
(3) Ava is a knight
(4) Daniel is a knight </think>
<answer> (1) Aurora is a knight
(2) Lily is a knave
(3) Ava is a knight
(4) Daniel is a knight </answer>
Aurora Lily Ava Daniel
(1) Aurora is a knight
(2) Lily is a knave
(3) Ava is a knight
(4) Daniel is a knight
(('and', ('telling-truth', 3), ('telling-truth', 2)), ('<=>', ('telling-truth', 2), ('lying', 0)), ('->', ('lying', 3), ('lying', 1)), ('or', ('telling-truth', 2), ('lying', 1)))
rule