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