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 5 inhabitants: Abigail, Samuel, Lily, Ava, and Mia. Abigail noted, "Samuel is a knight". Samuel stated, "Ava is a knight if and only if Abigail is a knight". "Mia is a knight" - Lily. In Ava's words: "Lily is a knight if and only if Ava is a knight". Mia noted, "Abigail 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 Abigail is a knight.  No contradiction is found in their claim that Samuel is a knight.
 Assume Samuel is a knight.  No contradiction is found in their claim that Ava is a knight if and only if Abigail is a knight.
 Assume Ava is a knight.  No contradiction is found in their claim that Lily is a knight if and only if Ava is a knight.
 Assume Lily is a knight.  No contradiction is found in their claim that Mia is a knight.
 Mia cannot be a knight, because this would contradict the claim of their own that Abigail is a knave.
 Mia cannot be a knave, because this would contradict the claim of Lily that Mia is a knight.
 We have exhausted all possibilities for Mia, so let us go back and reconsider Lily.
 Lily cannot be a knave, because this would contradict the claim of Ava that Lily is a knight if and only if Ava is a knight.
 We have exhausted all possibilities for Lily, so let us go back and reconsider Ava.
 Ava cannot be a knave, because this would contradict the claim of Samuel that Ava is a knight if and only if Abigail is a knight.
 We have exhausted all possibilities for Ava, so let us go back and reconsider Samuel.
 Samuel cannot be a knave, because this would contradict the claim of Abigail that Samuel is a knight.
 We have exhausted all possibilities for Samuel, so let us go back and reconsider Abigail.
 Assume Abigail is a knave.  No contradiction is found in their false claim that Samuel is a knight.
 Samuel cannot be a knight, because this would contradict the false claim of Abigail that Samuel is a knight.
 Assume Samuel is a knave.  No contradiction is found in their false claim that Ava is a knight if and only if Abigail is a knight.
 Assume Ava is a knight.  No contradiction is found in their claim that Lily is a knight if and only if Ava is a knight.
 Assume Lily is a knight.  No contradiction is found in their claim that Mia is a knight.
 Assume Mia is a knight.  No contradiction is found in their claim that Abigail is a knave.
 Therefore, (1) Abigail is a knave
 (2) Samuel is a knave
 (3) Lily is a knight
 (4) Ava is a knight
 (5) Mia 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 5 inhabitants: Abigail, Samuel, Lily, Ava, and Mia. Abigail noted, "Samuel is a knight". Samuel stated, "Ava is a knight if and only if Abigail is a knight". "Mia is a knight" - Lily. In Ava's words: "Lily is a knight if and only if Ava is a knight". Mia noted, "Abigail 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 Abigail is a knight. No contradiction is found in their claim that Samuel is a knight.
Assume Samuel is a knight. No contradiction is found in their claim that Ava is a knight if and only if Abigail is a knight.
Assume Ava is a knight. No contradiction is found in their claim that Lily is a knight if and only if Ava is a knight.
Assume Lily is a knight. No contradiction is found in their claim that Mia is a knight.
Mia cannot be a knight, because this would contradict the claim of their own that Abigail is a knave.
Mia cannot be a knave, because this would contradict the claim of Lily that Mia is a knight.
We have exhausted all possibilities for Mia, so let us go back and reconsider Lily.
Lily cannot be a knave, because this would contradict the claim of Ava that Lily is a knight if and only if Ava is a knight.
We have exhausted all possibilities for Lily, so let us go back and reconsider Ava.
Ava cannot be a knave, because this would contradict the claim of Samuel that Ava is a knight if and only if Abigail is a knight.
We have exhausted all possibilities for Ava, so let us go back and reconsider Samuel.
Samuel cannot be a knave, because this would contradict the claim of Abigail that Samuel is a knight.
We have exhausted all possibilities for Samuel, so let us go back and reconsider Abigail.
Assume Abigail is a knave. No contradiction is found in their false claim that Samuel is a knight.
Samuel cannot be a knight, because this would contradict the false claim of Abigail that Samuel is a knight.
Assume Samuel is a knave. No contradiction is found in their false claim that Ava is a knight if and only if Abigail is a knight.
Assume Ava is a knight. No contradiction is found in their claim that Lily is a knight if and only if Ava is a knight.
Assume Lily is a knight. No contradiction is found in their claim that Mia is a knight.
Assume Mia is a knight. No contradiction is found in their claim that Abigail is a knave.
Therefore, (1) Abigail is a knave
(2) Samuel is a knave
(3) Lily is a knight
(4) Ava is a knight
(5) Mia is a knight </think>
<answer> (1) Abigail is a knave
(2) Samuel is a knave
(3) Lily is a knight
(4) Ava is a knight
(5) Mia is a knight </answer>
Abigail Samuel Lily Ava Mia
(1) Abigail is a knave
(2) Samuel is a knave
(3) Lily is a knight
(4) Ava is a knight
(5) Mia is a knight
(('telling-truth', 1), ('<=>', ('telling-truth', 3), ('telling-truth', 0)), ('telling-truth', 4), ('<=>', ('telling-truth', 2), ('telling-truth', 3)), ('lying', 0))
rule