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: Matthew, Harper, Grace, Mason, Elizabeth, and Emily. Matthew noted, "Harper is a knave". Harper remarked, "Elizabeth is a knave if and only if Grace is a knight". In a statement by Grace: "Matthew is a knave or Elizabeth is a knave". In a statement by Mason: "Grace is not a knight". Elizabeth noted, "Grace is a knave or Emily is a knight". Emily was heard saying, "Matthew is a knight if and only if Matthew 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 Matthew is a knight.  No contradiction is found in their claim that Harper is a knave.
 Harper cannot be a knight, because this would contradict the claim of Matthew that Harper is a knave.
 Assume Harper is a knave.  No contradiction is found in their false claim that Elizabeth is a knave if and only if Grace is a knight.
 Assume Elizabeth is a knight.  No contradiction is found in their claim that Grace is a knave or Emily is a knight.
 Grace cannot be a knight, because this would contradict the claim of their own that Matthew is a knave or Elizabeth is a knave.
 Grace cannot be a knave, because this would contradict the false claim of Harper that Elizabeth is a knave if and only if Grace is a knight.
 We have exhausted all possibilities for Grace, so let us go back and reconsider Elizabeth.
 Assume Elizabeth is a knave.  No contradiction is found in their false claim that Grace is a knave or Emily is a knight.
 Grace cannot be a knight, because this would contradict the false claim of Harper that Elizabeth is a knave if and only if Grace is a knight.
 Grace cannot be a knave, because this would contradict the false claim of their own that Matthew is a knave or Elizabeth is a knave.
 We have exhausted all possibilities for Grace, Elizabeth, and Harper, so let us go back and reconsider Matthew.
 Assume Matthew is a knave.  No contradiction is found in their false claim that Harper is a knave.
 Assume Harper is a knight.  No contradiction is found in their claim that Elizabeth is a knave if and only if Grace is a knight.
 Assume Elizabeth is a knight.  No contradiction is found in their claim that Grace is a knave or Emily is a knight.
 Grace cannot be a knight, because this would contradict the claim of Harper that Elizabeth is a knave if and only if Grace is a knight.
 Grace cannot be a knave, because this would contradict the false claim of their own that Matthew is a knave or Elizabeth is a knave.
 We have exhausted all possibilities for Grace, so let us go back and reconsider Elizabeth.
 Assume Elizabeth is a knave.  No contradiction is found in their false claim that Grace is a knave or Emily is a knight.
 Assume Grace is a knight.  No contradiction is found in their claim that Matthew is a knave or Elizabeth is a knave.
 Emily cannot be a knight, because this would contradict the false claim of Elizabeth that Grace is a knave or Emily is a knight.
 Assume Emily is a knave.  No contradiction is found in their false claim that Matthew is a knight if and only if Matthew is a knave.
 Mason cannot be a knight, because this would contradict the claim of their own that Grace is not a knight.
 Assume Mason is a knave.  No contradiction is found in their false claim that Grace is not a knight.
 Therefore, (1) Matthew is a knave
 (2) Harper is a knight
 (3) Grace is a knight
 (4) Mason is a knave
 (5) Elizabeth is a knave
 (6) Emily 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: Matthew, Harper, Grace, Mason, Elizabeth, and Emily. Matthew noted, "Harper is a knave". Harper remarked, "Elizabeth is a knave if and only if Grace is a knight". In a statement by Grace: "Matthew is a knave or Elizabeth is a knave". In a statement by Mason: "Grace is not a knight". Elizabeth noted, "Grace is a knave or Emily is a knight". Emily was heard saying, "Matthew is a knight if and only if Matthew 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 Matthew is a knight. No contradiction is found in their claim that Harper is a knave.
Harper cannot be a knight, because this would contradict the claim of Matthew that Harper is a knave.
Assume Harper is a knave. No contradiction is found in their false claim that Elizabeth is a knave if and only if Grace is a knight.
Assume Elizabeth is a knight. No contradiction is found in their claim that Grace is a knave or Emily is a knight.
Grace cannot be a knight, because this would contradict the claim of their own that Matthew is a knave or Elizabeth is a knave.
Grace cannot be a knave, because this would contradict the false claim of Harper that Elizabeth is a knave if and only if Grace is a knight.
We have exhausted all possibilities for Grace, so let us go back and reconsider Elizabeth.
Assume Elizabeth is a knave. No contradiction is found in their false claim that Grace is a knave or Emily is a knight.
Grace cannot be a knight, because this would contradict the false claim of Harper that Elizabeth is a knave if and only if Grace is a knight.
Grace cannot be a knave, because this would contradict the false claim of their own that Matthew is a knave or Elizabeth is a knave.
We have exhausted all possibilities for Grace, Elizabeth, and Harper, so let us go back and reconsider Matthew.
Assume Matthew is a knave. No contradiction is found in their false claim that Harper is a knave.
Assume Harper is a knight. No contradiction is found in their claim that Elizabeth is a knave if and only if Grace is a knight.
Assume Elizabeth is a knight. No contradiction is found in their claim that Grace is a knave or Emily is a knight.
Grace cannot be a knight, because this would contradict the claim of Harper that Elizabeth is a knave if and only if Grace is a knight.
Grace cannot be a knave, because this would contradict the false claim of their own that Matthew is a knave or Elizabeth is a knave.
We have exhausted all possibilities for Grace, so let us go back and reconsider Elizabeth.
Assume Elizabeth is a knave. No contradiction is found in their false claim that Grace is a knave or Emily is a knight.
Assume Grace is a knight. No contradiction is found in their claim that Matthew is a knave or Elizabeth is a knave.
Emily cannot be a knight, because this would contradict the false claim of Elizabeth that Grace is a knave or Emily is a knight.
Assume Emily is a knave. No contradiction is found in their false claim that Matthew is a knight if and only if Matthew is a knave.
Mason cannot be a knight, because this would contradict the claim of their own that Grace is not a knight.
Assume Mason is a knave. No contradiction is found in their false claim that Grace is not a knight.
Therefore, (1) Matthew is a knave
(2) Harper is a knight
(3) Grace is a knight
(4) Mason is a knave
(5) Elizabeth is a knave
(6) Emily is a knave </think>
<answer> (1) Matthew is a knave
(2) Harper is a knight
(3) Grace is a knight
(4) Mason is a knave
(5) Elizabeth is a knave
(6) Emily is a knave </answer>
Matthew Harper Grace Mason Elizabeth Emily
(1) Matthew is a knave
(2) Harper is a knight
(3) Grace is a knight
(4) Mason is a knave
(5) Elizabeth is a knave
(6) Emily is a knave
(('lying', 1), ('<=>', ('lying', 4), ('telling-truth', 2)), ('or', ('lying', 0), ('lying', 4)), ('not', ('telling-truth', 2)), ('or', ('lying', 2), ('telling-truth', 5)), ('<=>', ('telling-truth', 0), ('lying', 0)))
rule