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 7 inhabitants: Aurora, Liam, Daniel, Owen, Jack, Penelope, and Avery. "Jack is a knave and Daniel is a knave," Aurora declared. Liam expressed that Penelope is a knave. "Aurora is a knight," Daniel claimed. Owen expressed that Jack is a knight. Jack expressed that If Penelope is a knight then Liam is a knave. Penelope said that Avery is a knight. "Jack is not a knight," Avery 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 Aurora is a knight.  No contradiction is found in their claim that Jack is a knave and Daniel is a knave.
 Jack cannot be a knight, because this would contradict the claim of Aurora that Jack is a knave and Daniel is a knave.
 Assume Jack is a knave.  No contradiction is found in their false claim that If Penelope is a knight then Liam is a knave.
 Assume Penelope is a knight.  No contradiction is found in their claim that Avery is a knight.
 Assume Avery is a knight.  No contradiction is found in their claim that Jack is not a knight.
 Liam cannot be a knight, because this would contradict the claim of their own that Penelope is a knave.
 Liam cannot be a knave, because this would contradict the false claim of Jack that If Penelope is a knight then Liam is a knave.
 We have exhausted all possibilities for Liam, so let us go back and reconsider Avery.
 Avery cannot be a knave, because this would contradict the claim of Penelope that Avery is a knight.
 We have exhausted all possibilities for Avery, so let us go back and reconsider Penelope.
 Penelope cannot be a knave, because this would contradict the false claim of Jack that If Penelope is a knight then Liam is a knave.
 We have exhausted all possibilities for Penelope and Jack, so let us go back and reconsider Aurora.
 Assume Aurora is a knave.  No contradiction is found in their false claim that Jack is a knave and Daniel is a knave.
 Assume Jack is a knight.  No contradiction is found in their claim that If Penelope is a knight then Liam is a knave.
 Assume Penelope is a knight.  No contradiction is found in their claim that Avery is a knight.
 Avery cannot be a knight, because this would contradict the claim of their own that Jack is not a knight.
 Avery cannot be a knave, because this would contradict the claim of Penelope that Avery is a knight.
 We have exhausted all possibilities for Avery, so let us go back and reconsider Penelope.
 Assume Penelope is a knave.  No contradiction is found in their false claim that Avery is a knight.
 Avery cannot be a knight, because this would contradict the false claim of Penelope that Avery is a knight.
 Assume Avery is a knave.  No contradiction is found in their false claim that Jack is not a knight.
 Assume Liam is a knight.  No contradiction is found in their claim that Penelope is a knave.
 Daniel cannot be a knight, because this would contradict the claim of their own that Aurora is a knight.
 Assume Daniel is a knave.  No contradiction is found in their false claim that Aurora is a knight.
 Assume Owen is a knight.  No contradiction is found in their claim that Jack is a knight.
 Therefore, (1) Aurora is a knave
 (2) Liam is a knight
 (3) Daniel is a knave
 (4) Owen is a knight
 (5) Jack is a knight
 (6) Penelope is a knave
 (7) Avery is a knave
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 7 inhabitants: Aurora, Liam, Daniel, Owen, Jack, Penelope, and Avery. "Jack is a knave and Daniel is a knave," Aurora declared. Liam expressed that Penelope is a knave. "Aurora is a knight," Daniel claimed. Owen expressed that Jack is a knight. Jack expressed that If Penelope is a knight then Liam is a knave. Penelope said that Avery is a knight. "Jack is not a knight," Avery 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 Aurora is a knight. No contradiction is found in their claim that Jack is a knave and Daniel is a knave.
Jack cannot be a knight, because this would contradict the claim of Aurora that Jack is a knave and Daniel is a knave.
Assume Jack is a knave. No contradiction is found in their false claim that If Penelope is a knight then Liam is a knave.
Assume Penelope is a knight. No contradiction is found in their claim that Avery is a knight.
Assume Avery is a knight. No contradiction is found in their claim that Jack is not a knight.
Liam cannot be a knight, because this would contradict the claim of their own that Penelope is a knave.
Liam cannot be a knave, because this would contradict the false claim of Jack that If Penelope is a knight then Liam is a knave.
We have exhausted all possibilities for Liam, so let us go back and reconsider Avery.
Avery cannot be a knave, because this would contradict the claim of Penelope that Avery is a knight.
We have exhausted all possibilities for Avery, so let us go back and reconsider Penelope.
Penelope cannot be a knave, because this would contradict the false claim of Jack that If Penelope is a knight then Liam is a knave.
We have exhausted all possibilities for Penelope and Jack, so let us go back and reconsider Aurora.
Assume Aurora is a knave. No contradiction is found in their false claim that Jack is a knave and Daniel is a knave.
Assume Jack is a knight. No contradiction is found in their claim that If Penelope is a knight then Liam is a knave.
Assume Penelope is a knight. No contradiction is found in their claim that Avery is a knight.
Avery cannot be a knight, because this would contradict the claim of their own that Jack is not a knight.
Avery cannot be a knave, because this would contradict the claim of Penelope that Avery is a knight.
We have exhausted all possibilities for Avery, so let us go back and reconsider Penelope.
Assume Penelope is a knave. No contradiction is found in their false claim that Avery is a knight.
Avery cannot be a knight, because this would contradict the false claim of Penelope that Avery is a knight.
Assume Avery is a knave. No contradiction is found in their false claim that Jack is not a knight.
Assume Liam is a knight. No contradiction is found in their claim that Penelope is a knave.
Daniel cannot be a knight, because this would contradict the claim of their own that Aurora is a knight.
Assume Daniel is a knave. No contradiction is found in their false claim that Aurora is a knight.
Assume Owen is a knight. No contradiction is found in their claim that Jack is a knight.
Therefore, (1) Aurora is a knave
(2) Liam is a knight
(3) Daniel is a knave
(4) Owen is a knight
(5) Jack is a knight
(6) Penelope is a knave
(7) Avery is a knave </think>
<answer> (1) Aurora is a knave
(2) Liam is a knight
(3) Daniel is a knave
(4) Owen is a knight
(5) Jack is a knight
(6) Penelope is a knave
(7) Avery is a knave </answer>
Aurora Liam Daniel Owen Jack Penelope Avery
(1) Aurora is a knave
(2) Liam is a knight
(3) Daniel is a knave
(4) Owen is a knight
(5) Jack is a knight
(6) Penelope is a knave
(7) Avery is a knave
(('and', ('lying', 4), ('lying', 2)), ('lying', 5), ('telling-truth', 0), ('telling-truth', 4), ('->', ('telling-truth', 5), ('lying', 1)), ('telling-truth', 6), ('not', ('telling-truth', 4)))
rule