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: Avery, Luke, Lucas, and Daniel. In Avery's words: "If Lucas is a knave then Daniel is a knave". Luke said that Daniel is a knave or Avery is a knave. In Lucas's words: "Luke is a knave if and only if Luke is a knight". Daniel asserted: "Avery is a knight or Avery 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 Avery is a knight.  No contradiction is found in their claim that If Lucas is a knave then Daniel is a knave.
 Lucas cannot be a knight, because this would contradict the claim of their own that Luke is a knave if and only if Luke is a knight.
 Assume Lucas is a knave.  No contradiction is found in their false claim that Luke is a knave if and only if Luke is a knight.
 Assume Luke is a knight.  No contradiction is found in their claim that Daniel is a knave or Avery is a knave.
 Daniel cannot be a knight, because this would contradict the claim of Avery that If Lucas is a knave then Daniel is a knave.
 Daniel cannot be a knave, because this would contradict the false claim of their own that Avery is a knight or Avery is a knave.
 We have exhausted all possibilities for Daniel, so let us go back and reconsider Luke.
 Assume Luke is a knave.  No contradiction is found in their false claim that Daniel is a knave or Avery is a knave.
 Daniel cannot be a knight, because this would contradict the claim of Avery that If Lucas is a knave then Daniel is a knave.
 Daniel cannot be a knave, because this would contradict the false claim of Luke that Daniel is a knave or Avery is a knave.
 We have exhausted all possibilities for Daniel, Luke, and Lucas, so let us go back and reconsider Avery.
 Assume Avery is a knave.  No contradiction is found in their false claim that If Lucas is a knave then Daniel is a knave.
 Lucas cannot be a knight, because this would contradict the false claim of Avery that If Lucas is a knave then Daniel is a knave.
 Assume Lucas is a knave.  No contradiction is found in their false claim that Luke is a knave if and only if Luke is a knight.
 Assume Luke is a knight.  No contradiction is found in their claim that Daniel is a knave or Avery is a knave.
 Assume Daniel is a knight.  No contradiction is found in their claim that Avery is a knight or Avery is a knave.
 Therefore, (1) Avery is a knave
 (2) Luke is a knight
 (3) Lucas is a knave
 (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: Avery, Luke, Lucas, and Daniel. In Avery's words: "If Lucas is a knave then Daniel is a knave". Luke said that Daniel is a knave or Avery is a knave. In Lucas's words: "Luke is a knave if and only if Luke is a knight". Daniel asserted: "Avery is a knight or Avery 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 Avery is a knight. No contradiction is found in their claim that If Lucas is a knave then Daniel is a knave.
Lucas cannot be a knight, because this would contradict the claim of their own that Luke is a knave if and only if Luke is a knight.
Assume Lucas is a knave. No contradiction is found in their false claim that Luke is a knave if and only if Luke is a knight.
Assume Luke is a knight. No contradiction is found in their claim that Daniel is a knave or Avery is a knave.
Daniel cannot be a knight, because this would contradict the claim of Avery that If Lucas is a knave then Daniel is a knave.
Daniel cannot be a knave, because this would contradict the false claim of their own that Avery is a knight or Avery is a knave.
We have exhausted all possibilities for Daniel, so let us go back and reconsider Luke.
Assume Luke is a knave. No contradiction is found in their false claim that Daniel is a knave or Avery is a knave.
Daniel cannot be a knight, because this would contradict the claim of Avery that If Lucas is a knave then Daniel is a knave.
Daniel cannot be a knave, because this would contradict the false claim of Luke that Daniel is a knave or Avery is a knave.
We have exhausted all possibilities for Daniel, Luke, and Lucas, so let us go back and reconsider Avery.
Assume Avery is a knave. No contradiction is found in their false claim that If Lucas is a knave then Daniel is a knave.
Lucas cannot be a knight, because this would contradict the false claim of Avery that If Lucas is a knave then Daniel is a knave.
Assume Lucas is a knave. No contradiction is found in their false claim that Luke is a knave if and only if Luke is a knight.
Assume Luke is a knight. No contradiction is found in their claim that Daniel is a knave or Avery is a knave.
Assume Daniel is a knight. No contradiction is found in their claim that Avery is a knight or Avery is a knave.
Therefore, (1) Avery is a knave
(2) Luke is a knight
(3) Lucas is a knave
(4) Daniel is a knight </think>
<answer> (1) Avery is a knave
(2) Luke is a knight
(3) Lucas is a knave
(4) Daniel is a knight </answer>
Avery Luke Lucas Daniel
(1) Avery is a knave
(2) Luke is a knight
(3) Lucas is a knave
(4) Daniel is a knight
(('->', ('lying', 2), ('lying', 3)), ('or', ('lying', 3), ('lying', 0)), ('<=>', ('lying', 1), ('telling-truth', 1)), ('or', ('telling-truth', 0), ('lying', 0)))
rule