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 6 inhabitants: Luke, Grace, Olivia, Evelyn, Elizabeth, and Ava. "Ava is a knight and Evelyn is a knight," Luke claimed. Grace stated, "Olivia is a knave and Elizabeth is a knave". Olivia was heard saying, "Luke is a knave if and only if Grace is a knight". Evelyn told you that Elizabeth is not a knight. "Olivia is a knave if and only if Elizabeth is a knight," Elizabeth mentioned. "Evelyn is a knave if and only if Olivia is a knave," Ava declared. 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 Luke is a knight.  No contradiction is found in their claim that Ava is a knight and Evelyn is a knight.
 Assume Ava is a knight.  No contradiction is found in their claim that Evelyn is a knave if and only if Olivia is a knave.
 Assume Evelyn is a knight.  No contradiction is found in their claim that Elizabeth is not a knight.
 Elizabeth cannot be a knight, because this would contradict the claim of Evelyn that Elizabeth is not a knight.
 Assume Elizabeth is a knave.  No contradiction is found in their false claim that Olivia is a knave if and only if Elizabeth is a knight.
 Olivia cannot be a knight, because this would contradict the false claim of Elizabeth that Olivia is a knave if and only if Elizabeth is a knight.
 Olivia cannot be a knave, because this would contradict the claim of Ava that Evelyn is a knave if and only if Olivia is a knave.
 We have exhausted all possibilities for Olivia and Elizabeth, so let us go back and reconsider Evelyn.
 Evelyn cannot be a knave, because this would contradict the claim of Luke that Ava is a knight and Evelyn is a knight.
 We have exhausted all possibilities for Evelyn, so let us go back and reconsider Ava.
 Ava cannot be a knave, because this would contradict the claim of Luke that Ava is a knight and Evelyn is a knight.
 We have exhausted all possibilities for Ava, so let us go back and reconsider Luke.
 Assume Luke is a knave.  No contradiction is found in their false claim that Ava is a knight and Evelyn is a knight.
 Assume Ava is a knight.  No contradiction is found in their claim that Evelyn is a knave if and only if Olivia is a knave.
 Evelyn cannot be a knight, because this would contradict the false claim of Luke that Ava is a knight and Evelyn is a knight.
 Assume Evelyn is a knave.  No contradiction is found in their false claim that Elizabeth is not a knight.
 Assume Elizabeth is a knight.  No contradiction is found in their claim that Olivia is a knave if and only if Elizabeth is a knight.
 Olivia cannot be a knight, because this would contradict the claim of Elizabeth that Olivia is a knave if and only if Elizabeth is a knight.
 Assume Olivia is a knave.  No contradiction is found in their false claim that Luke is a knave if and only if Grace is a knight.
 Grace cannot be a knight, because this would contradict the claim of their own that Olivia is a knave and Elizabeth is a knave.
 Assume Grace is a knave.  No contradiction is found in their false claim that Olivia is a knave and Elizabeth is a knave.
 Therefore, (1) Luke is a knave
 (2) Grace is a knave
 (3) Olivia is a knave
 (4) Evelyn is a knave
 (5) Elizabeth is a knight
 (6) Ava 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 6 inhabitants: Luke, Grace, Olivia, Evelyn, Elizabeth, and Ava. "Ava is a knight and Evelyn is a knight," Luke claimed. Grace stated, "Olivia is a knave and Elizabeth is a knave". Olivia was heard saying, "Luke is a knave if and only if Grace is a knight". Evelyn told you that Elizabeth is not a knight. "Olivia is a knave if and only if Elizabeth is a knight," Elizabeth mentioned. "Evelyn is a knave if and only if Olivia is a knave," Ava declared. 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 Luke is a knight. No contradiction is found in their claim that Ava is a knight and Evelyn is a knight.
Assume Ava is a knight. No contradiction is found in their claim that Evelyn is a knave if and only if Olivia is a knave.
Assume Evelyn is a knight. No contradiction is found in their claim that Elizabeth is not a knight.
Elizabeth cannot be a knight, because this would contradict the claim of Evelyn that Elizabeth is not a knight.
Assume Elizabeth is a knave. No contradiction is found in their false claim that Olivia is a knave if and only if Elizabeth is a knight.
Olivia cannot be a knight, because this would contradict the false claim of Elizabeth that Olivia is a knave if and only if Elizabeth is a knight.
Olivia cannot be a knave, because this would contradict the claim of Ava that Evelyn is a knave if and only if Olivia is a knave.
We have exhausted all possibilities for Olivia and Elizabeth, so let us go back and reconsider Evelyn.
Evelyn cannot be a knave, because this would contradict the claim of Luke that Ava is a knight and Evelyn is a knight.
We have exhausted all possibilities for Evelyn, so let us go back and reconsider Ava.
Ava cannot be a knave, because this would contradict the claim of Luke that Ava is a knight and Evelyn is a knight.
We have exhausted all possibilities for Ava, so let us go back and reconsider Luke.
Assume Luke is a knave. No contradiction is found in their false claim that Ava is a knight and Evelyn is a knight.
Assume Ava is a knight. No contradiction is found in their claim that Evelyn is a knave if and only if Olivia is a knave.
Evelyn cannot be a knight, because this would contradict the false claim of Luke that Ava is a knight and Evelyn is a knight.
Assume Evelyn is a knave. No contradiction is found in their false claim that Elizabeth is not a knight.
Assume Elizabeth is a knight. No contradiction is found in their claim that Olivia is a knave if and only if Elizabeth is a knight.
Olivia cannot be a knight, because this would contradict the claim of Elizabeth that Olivia is a knave if and only if Elizabeth is a knight.
Assume Olivia is a knave. No contradiction is found in their false claim that Luke is a knave if and only if Grace is a knight.
Grace cannot be a knight, because this would contradict the claim of their own that Olivia is a knave and Elizabeth is a knave.
Assume Grace is a knave. No contradiction is found in their false claim that Olivia is a knave and Elizabeth is a knave.
Therefore, (1) Luke is a knave
(2) Grace is a knave
(3) Olivia is a knave
(4) Evelyn is a knave
(5) Elizabeth is a knight
(6) Ava is a knight </think>
<answer> (1) Luke is a knave
(2) Grace is a knave
(3) Olivia is a knave
(4) Evelyn is a knave
(5) Elizabeth is a knight
(6) Ava is a knight </answer>
Luke Grace Olivia Evelyn Elizabeth Ava
(1) Luke is a knave
(2) Grace is a knave
(3) Olivia is a knave
(4) Evelyn is a knave
(5) Elizabeth is a knight
(6) Ava is a knight
(('and', ('telling-truth', 5), ('telling-truth', 3)), ('and', ('lying', 2), ('lying', 4)), ('<=>', ('lying', 0), ('telling-truth', 1)), ('not', ('telling-truth', 4)), ('<=>', ('lying', 2), ('telling-truth', 4)), ('<=>', ('lying', 3), ('lying', 2)))
rule