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