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: Logan, Lucas, Elizabeth, Grace, Evelyn, and Emma. Logan said that Emma is a knave and Emma is a knight. Lucas noted, "Grace is a knave if and only if Lucas is a knight". Elizabeth told you that Lucas is a knight if and only if Grace is a knight. Grace said, "Elizabeth is a knave." Evelyn expressed that Elizabeth is a knave and Emma is a knight. According to Emma, "Evelyn is a knight if and only if Logan 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
Logan cannot be a knight, because this would contradict the claim of their own that Emma is a knave and Emma is a knight.
 Assume Logan is a knave.  No contradiction is found in their false claim that Emma is a knave and Emma is a knight.
 Assume Emma is a knight.  No contradiction is found in their claim that Evelyn is a knight if and only if Logan is a knave.
 Assume Evelyn is a knight.  No contradiction is found in their claim that Elizabeth is a knave and Emma is a knight.
 Elizabeth cannot be a knight, because this would contradict the claim of Evelyn that Elizabeth is a knave and Emma is a knight.
 Assume Elizabeth is a knave.  No contradiction is found in their false claim that Lucas is a knight if and only if Grace is a knight.
 Assume Lucas is a knight.  No contradiction is found in their claim that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knight, because this would contradict the claim of Lucas that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knave, because this would contradict the false claim of their own that Elizabeth is a knave.
 We have exhausted all possibilities for Grace, so let us go back and reconsider Lucas.
 Assume Lucas is a knave.  No contradiction is found in their false claim that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knight, because this would contradict the false claim of Lucas that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knave, because this would contradict the false claim of Elizabeth that Lucas is a knight if and only if Grace is a knight.
 We have exhausted all possibilities for Grace, Lucas, and Elizabeth, so let us go back and reconsider Evelyn.
 Evelyn cannot be a knave, because this would contradict the claim of Emma that Evelyn is a knight if and only if Logan is a knave.
 We have exhausted all possibilities for Evelyn, so let us go back and reconsider Emma.
 Assume Emma is a knave.  No contradiction is found in their false claim that Evelyn is a knight if and only if Logan is a knave.
 Evelyn cannot be a knight, because this would contradict the claim of their own that Elizabeth is a knave and Emma is a knight.
 Assume Evelyn is a knave.  No contradiction is found in their false claim that Elizabeth is a knave and Emma is a knight.
 Assume Elizabeth is a knight.  No contradiction is found in their claim that Lucas is a knight if and only if Grace is a knight.
 Assume Lucas is a knight.  No contradiction is found in their claim that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knight, because this would contradict the claim of Lucas that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knave, because this would contradict the claim of Elizabeth that Lucas is a knight if and only if Grace is a knight.
 We have exhausted all possibilities for Grace, so let us go back and reconsider Lucas.
 Assume Lucas is a knave.  No contradiction is found in their false claim that Grace is a knave if and only if Lucas is a knight.
 Grace cannot be a knight, because this would contradict the false claim of Lucas that Grace is a knave if and only if Lucas is a knight.
 Assume Grace is a knave.  No contradiction is found in their false claim that Elizabeth is a knave.
 Therefore, (1) Logan is a knave
 (2) Lucas is a knave
 (3) Elizabeth is a knight
 (4) Grace is a knave
 (5) Evelyn is a knave
 (6) Emma 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 6 inhabitants: Logan, Lucas, Elizabeth, Grace, Evelyn, and Emma. Logan said that Emma is a knave and Emma is a knight. Lucas noted, "Grace is a knave if and only if Lucas is a knight". Elizabeth told you that Lucas is a knight if and only if Grace is a knight. Grace said, "Elizabeth is a knave." Evelyn expressed that Elizabeth is a knave and Emma is a knight. According to Emma, "Evelyn is a knight if and only if Logan 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> 
Logan cannot be a knight, because this would contradict the claim of their own that Emma is a knave and Emma is a knight.
Assume Logan is a knave. No contradiction is found in their false claim that Emma is a knave and Emma is a knight.
Assume Emma is a knight. No contradiction is found in their claim that Evelyn is a knight if and only if Logan is a knave.
Assume Evelyn is a knight. No contradiction is found in their claim that Elizabeth is a knave and Emma is a knight.
Elizabeth cannot be a knight, because this would contradict the claim of Evelyn that Elizabeth is a knave and Emma is a knight.
Assume Elizabeth is a knave. No contradiction is found in their false claim that Lucas is a knight if and only if Grace is a knight.
Assume Lucas is a knight. No contradiction is found in their claim that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knight, because this would contradict the claim of Lucas that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knave, because this would contradict the false claim of their own that Elizabeth is a knave.
We have exhausted all possibilities for Grace, so let us go back and reconsider Lucas.
Assume Lucas is a knave. No contradiction is found in their false claim that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knight, because this would contradict the false claim of Lucas that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knave, because this would contradict the false claim of Elizabeth that Lucas is a knight if and only if Grace is a knight.
We have exhausted all possibilities for Grace, Lucas, and Elizabeth, so let us go back and reconsider Evelyn.
Evelyn cannot be a knave, because this would contradict the claim of Emma that Evelyn is a knight if and only if Logan is a knave.
We have exhausted all possibilities for Evelyn, so let us go back and reconsider Emma.
Assume Emma is a knave. No contradiction is found in their false claim that Evelyn is a knight if and only if Logan is a knave.
Evelyn cannot be a knight, because this would contradict the claim of their own that Elizabeth is a knave and Emma is a knight.
Assume Evelyn is a knave. No contradiction is found in their false claim that Elizabeth is a knave and Emma is a knight.
Assume Elizabeth is a knight. No contradiction is found in their claim that Lucas is a knight if and only if Grace is a knight.
Assume Lucas is a knight. No contradiction is found in their claim that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knight, because this would contradict the claim of Lucas that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knave, because this would contradict the claim of Elizabeth that Lucas is a knight if and only if Grace is a knight.
We have exhausted all possibilities for Grace, so let us go back and reconsider Lucas.
Assume Lucas is a knave. No contradiction is found in their false claim that Grace is a knave if and only if Lucas is a knight.
Grace cannot be a knight, because this would contradict the false claim of Lucas that Grace is a knave if and only if Lucas is a knight.
Assume Grace is a knave. No contradiction is found in their false claim that Elizabeth is a knave.
Therefore, (1) Logan is a knave
(2) Lucas is a knave
(3) Elizabeth is a knight
(4) Grace is a knave
(5) Evelyn is a knave
(6) Emma is a knave </think>
<answer> (1) Logan is a knave
(2) Lucas is a knave
(3) Elizabeth is a knight
(4) Grace is a knave
(5) Evelyn is a knave
(6) Emma is a knave </answer>
Logan Lucas Elizabeth Grace Evelyn Emma
(1) Logan is a knave
(2) Lucas is a knave
(3) Elizabeth is a knight
(4) Grace is a knave
(5) Evelyn is a knave
(6) Emma is a knave
(('and', ('lying', 5), ('telling-truth', 5)), ('<=>', ('lying', 3), ('telling-truth', 1)), ('<=>', ('telling-truth', 1), ('telling-truth', 3)), ('lying', 2), ('and', ('lying', 2), ('telling-truth', 5)), ('<=>', ('telling-truth', 4), ('lying', 0)))
rule