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: Grace, Owen, Matthew, Ethan, Emily, and Oliver. Grace remarked, "Ethan is a knave". Owen expressed that Matthew is a knight and Oliver is a knight. As Matthew put it, "Oliver is a knave and Grace is a knave". Ethan commented, "Emily is a knight and Matthew is a knight". "If Grace is a knight then Matthew is a knight" - Emily. Oliver remarked, "Emily is a knight if and only if Owen 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 Grace is a knight.  No contradiction is found in their claim that Ethan is a knave.
 Ethan cannot be a knight, because this would contradict the claim of Grace that Ethan is a knave.
 Assume Ethan is a knave.  No contradiction is found in their false claim that Emily is a knight and Matthew is a knight.
 Assume Emily is a knight.  No contradiction is found in their claim that If Grace is a knight then Matthew is a knight.
 Matthew cannot be a knight, because this would contradict the claim of their own that Oliver is a knave and Grace is a knave.
 Matthew cannot be a knave, because this would contradict the claim of Emily that If Grace is a knight then Matthew is a knight.
 We have exhausted all possibilities for Matthew, so let us go back and reconsider Emily.
 Assume Emily is a knave.  No contradiction is found in their false claim that If Grace is a knight then Matthew is a knight.
 Matthew cannot be a knight, because this would contradict the claim of their own that Oliver is a knave and Grace is a knave.
 Assume Matthew is a knave.  No contradiction is found in their false claim that Oliver is a knave and Grace is a knave.
 Assume Oliver is a knight.  No contradiction is found in their claim that Emily is a knight if and only if Owen is a knave.
 Owen cannot be a knight, because this would contradict the claim of their own that Matthew is a knight and Oliver is a knight.
 Owen cannot be a knave, because this would contradict the claim of Oliver that Emily is a knight if and only if Owen is a knave.
 We have exhausted all possibilities for Owen, so let us go back and reconsider Oliver.
 Assume Oliver is a knave.  No contradiction is found in their false claim that Emily is a knight if and only if Owen is a knave.
 Owen cannot be a knight, because this would contradict the claim of their own that Matthew is a knight and Oliver is a knight.
 Assume Owen is a knave.  No contradiction is found in their false claim that Matthew is a knight and Oliver is a knight.
 Therefore, (1) Grace is a knight
 (2) Owen is a knave
 (3) Matthew is a knave
 (4) Ethan is a knave
 (5) Emily is a knave
 (6) Oliver 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: Grace, Owen, Matthew, Ethan, Emily, and Oliver. Grace remarked, "Ethan is a knave". Owen expressed that Matthew is a knight and Oliver is a knight. As Matthew put it, "Oliver is a knave and Grace is a knave". Ethan commented, "Emily is a knight and Matthew is a knight". "If Grace is a knight then Matthew is a knight" - Emily. Oliver remarked, "Emily is a knight if and only if Owen 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 Grace is a knight. No contradiction is found in their claim that Ethan is a knave.
Ethan cannot be a knight, because this would contradict the claim of Grace that Ethan is a knave.
Assume Ethan is a knave. No contradiction is found in their false claim that Emily is a knight and Matthew is a knight.
Assume Emily is a knight. No contradiction is found in their claim that If Grace is a knight then Matthew is a knight.
Matthew cannot be a knight, because this would contradict the claim of their own that Oliver is a knave and Grace is a knave.
Matthew cannot be a knave, because this would contradict the claim of Emily that If Grace is a knight then Matthew is a knight.
We have exhausted all possibilities for Matthew, so let us go back and reconsider Emily.
Assume Emily is a knave. No contradiction is found in their false claim that If Grace is a knight then Matthew is a knight.
Matthew cannot be a knight, because this would contradict the claim of their own that Oliver is a knave and Grace is a knave.
Assume Matthew is a knave. No contradiction is found in their false claim that Oliver is a knave and Grace is a knave.
Assume Oliver is a knight. No contradiction is found in their claim that Emily is a knight if and only if Owen is a knave.
Owen cannot be a knight, because this would contradict the claim of their own that Matthew is a knight and Oliver is a knight.
Owen cannot be a knave, because this would contradict the claim of Oliver that Emily is a knight if and only if Owen is a knave.
We have exhausted all possibilities for Owen, so let us go back and reconsider Oliver.
Assume Oliver is a knave. No contradiction is found in their false claim that Emily is a knight if and only if Owen is a knave.
Owen cannot be a knight, because this would contradict the claim of their own that Matthew is a knight and Oliver is a knight.
Assume Owen is a knave. No contradiction is found in their false claim that Matthew is a knight and Oliver is a knight.
Therefore, (1) Grace is a knight
(2) Owen is a knave
(3) Matthew is a knave
(4) Ethan is a knave
(5) Emily is a knave
(6) Oliver is a knave </think>
<answer> (1) Grace is a knight
(2) Owen is a knave
(3) Matthew is a knave
(4) Ethan is a knave
(5) Emily is a knave
(6) Oliver is a knave </answer>
Grace Owen Matthew Ethan Emily Oliver
(1) Grace is a knight
(2) Owen is a knave
(3) Matthew is a knave
(4) Ethan is a knave
(5) Emily is a knave
(6) Oliver is a knave
(('lying', 3), ('and', ('telling-truth', 2), ('telling-truth', 5)), ('and', ('lying', 5), ('lying', 0)), ('and', ('telling-truth', 4), ('telling-truth', 2)), ('->', ('telling-truth', 0), ('telling-truth', 2)), ('<=>', ('telling-truth', 4), ('lying', 1)))
rule