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: Abigail, Sophia, Matthew, Joseph, Emma, and Riley. "Sophia is a knight if and only if Sophia is a knave," Abigail claimed. Sophia noted, "Emma is a knave". Matthew noted, "Sophia is a knave if and only if Joseph is a knight". Joseph noted, "Sophia is a knight or Riley is a knave". "Riley is a knight and Riley is a knave" - Emma. "If Abigail is a knight then Emma is a knave," Riley mentioned. 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
Abigail cannot be a knight, because this would contradict the claim of their own that Sophia is a knight if and only if Sophia is a knave.
 Assume Abigail is a knave.  No contradiction is found in their false claim that Sophia is a knight if and only if Sophia is a knave.
 Assume Sophia is a knight.  No contradiction is found in their claim that Emma is a knave.
 Emma cannot be a knight, because this would contradict the claim of Sophia that Emma is a knave.
 Assume Emma is a knave.  No contradiction is found in their false claim that Riley is a knight and Riley is a knave.
 Assume Riley is a knight.  No contradiction is found in their claim that If Abigail is a knight then Emma is a knave.
 Assume Matthew is a knight.  No contradiction is found in their claim that Sophia is a knave if and only if Joseph is a knight.
 Joseph cannot be a knight, because this would contradict the claim of Matthew that Sophia is a knave if and only if Joseph is a knight.
 Joseph cannot be a knave, because this would contradict the false claim of their own that Sophia is a knight or Riley is a knave.
 We have exhausted all possibilities for Joseph, so let us go back and reconsider Matthew.
 Assume Matthew is a knave.  No contradiction is found in their false claim that Sophia is a knave if and only if Joseph is a knight.
 Assume Joseph is a knight.  No contradiction is found in their claim that Sophia is a knight or Riley is a knave.
 Therefore, (1) Abigail is a knave
 (2) Sophia is a knight
 (3) Matthew is a knave
 (4) Joseph is a knight
 (5) Emma is a knave
 (6) Riley 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: Abigail, Sophia, Matthew, Joseph, Emma, and Riley. "Sophia is a knight if and only if Sophia is a knave," Abigail claimed. Sophia noted, "Emma is a knave". Matthew noted, "Sophia is a knave if and only if Joseph is a knight". Joseph noted, "Sophia is a knight or Riley is a knave". "Riley is a knight and Riley is a knave" - Emma. "If Abigail is a knight then Emma is a knave," Riley mentioned. 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> 
Abigail cannot be a knight, because this would contradict the claim of their own that Sophia is a knight if and only if Sophia is a knave.
Assume Abigail is a knave. No contradiction is found in their false claim that Sophia is a knight if and only if Sophia is a knave.
Assume Sophia is a knight. No contradiction is found in their claim that Emma is a knave.
Emma cannot be a knight, because this would contradict the claim of Sophia that Emma is a knave.
Assume Emma is a knave. No contradiction is found in their false claim that Riley is a knight and Riley is a knave.
Assume Riley is a knight. No contradiction is found in their claim that If Abigail is a knight then Emma is a knave.
Assume Matthew is a knight. No contradiction is found in their claim that Sophia is a knave if and only if Joseph is a knight.
Joseph cannot be a knight, because this would contradict the claim of Matthew that Sophia is a knave if and only if Joseph is a knight.
Joseph cannot be a knave, because this would contradict the false claim of their own that Sophia is a knight or Riley is a knave.
We have exhausted all possibilities for Joseph, so let us go back and reconsider Matthew.
Assume Matthew is a knave. No contradiction is found in their false claim that Sophia is a knave if and only if Joseph is a knight.
Assume Joseph is a knight. No contradiction is found in their claim that Sophia is a knight or Riley is a knave.
Therefore, (1) Abigail is a knave
(2) Sophia is a knight
(3) Matthew is a knave
(4) Joseph is a knight
(5) Emma is a knave
(6) Riley is a knight </think>
<answer> (1) Abigail is a knave
(2) Sophia is a knight
(3) Matthew is a knave
(4) Joseph is a knight
(5) Emma is a knave
(6) Riley is a knight </answer>
Abigail Sophia Matthew Joseph Emma Riley
(1) Abigail is a knave
(2) Sophia is a knight
(3) Matthew is a knave
(4) Joseph is a knight
(5) Emma is a knave
(6) Riley is a knight
(('<=>', ('telling-truth', 1), ('lying', 1)), ('lying', 4), ('<=>', ('lying', 1), ('telling-truth', 3)), ('or', ('telling-truth', 1), ('lying', 5)), ('and', ('telling-truth', 5), ('lying', 5)), ('->', ('telling-truth', 0), ('lying', 4)))
rule