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: Aiden, Victoria, Lucas, Benjamin, and Grace. "Benjamin is a knight if and only if Grace is a knight," Aiden declared. Victoria was heard saying, "Aiden is a knight". Lucas said that Aiden is not a knight. Benjamin was heard saying, "Lucas is a knave". Grace noted, "Victoria 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 Aiden is a knight.  No contradiction is found in their claim that Benjamin is a knight if and only if Grace is a knight.
 Assume Benjamin is a knight.  No contradiction is found in their claim that Lucas is a knave.
 Lucas cannot be a knight, because this would contradict the claim of their own that Aiden is not a knight.
 Assume Lucas is a knave.  No contradiction is found in their false claim that Aiden is not a knight.
 Assume Grace is a knight.  No contradiction is found in their claim that Victoria is a knave.
 Victoria cannot be a knight, because this would contradict the claim of Grace that Victoria is a knave.
 Victoria cannot be a knave, because this would contradict the false claim of their own that Aiden is a knight.
 We have exhausted all possibilities for Victoria, so let us go back and reconsider Grace.
 Grace cannot be a knave, because this would contradict the claim of Aiden that Benjamin is a knight if and only if Grace is a knight.
 We have exhausted all possibilities for Grace and Lucas, so let us go back and reconsider Benjamin.
 Assume Benjamin is a knave.  No contradiction is found in their false claim that Lucas is a knave.
 Lucas cannot be a knight, because this would contradict the claim of their own that Aiden is not a knight.
 Lucas cannot be a knave, because this would contradict the false claim of Benjamin that Lucas is a knave.
 We have exhausted all possibilities for Lucas and Benjamin, so let us go back and reconsider Aiden.
 Assume Aiden is a knave.  No contradiction is found in their false claim that Benjamin is a knight if and only if Grace is a knight.
 Assume Benjamin is a knight.  No contradiction is found in their claim that Lucas is a knave.
 Lucas cannot be a knight, because this would contradict the claim of Benjamin that Lucas is a knave.
 Lucas cannot be a knave, because this would contradict the false claim of their own that Aiden is not a knight.
 We have exhausted all possibilities for Lucas, so let us go back and reconsider Benjamin.
 Assume Benjamin is a knave.  No contradiction is found in their false claim that Lucas is a knave.
 Assume Lucas is a knight.  No contradiction is found in their claim that Aiden is not a knight.
 Assume Grace is a knight.  No contradiction is found in their claim that Victoria is a knave.
 Victoria cannot be a knight, because this would contradict the claim of their own that Aiden is a knight.
 Assume Victoria is a knave.  No contradiction is found in their false claim that Aiden is a knight.
 Therefore, (1) Aiden is a knave
 (2) Victoria is a knave
 (3) Lucas is a knight
 (4) Benjamin is a knave
 (5) Grace 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: Aiden, Victoria, Lucas, Benjamin, and Grace. "Benjamin is a knight if and only if Grace is a knight," Aiden declared. Victoria was heard saying, "Aiden is a knight". Lucas said that Aiden is not a knight. Benjamin was heard saying, "Lucas is a knave". Grace noted, "Victoria 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 Aiden is a knight. No contradiction is found in their claim that Benjamin is a knight if and only if Grace is a knight.
Assume Benjamin is a knight. No contradiction is found in their claim that Lucas is a knave.
Lucas cannot be a knight, because this would contradict the claim of their own that Aiden is not a knight.
Assume Lucas is a knave. No contradiction is found in their false claim that Aiden is not a knight.
Assume Grace is a knight. No contradiction is found in their claim that Victoria is a knave.
Victoria cannot be a knight, because this would contradict the claim of Grace that Victoria is a knave.
Victoria cannot be a knave, because this would contradict the false claim of their own that Aiden is a knight.
We have exhausted all possibilities for Victoria, so let us go back and reconsider Grace.
Grace cannot be a knave, because this would contradict the claim of Aiden that Benjamin is a knight if and only if Grace is a knight.
We have exhausted all possibilities for Grace and Lucas, so let us go back and reconsider Benjamin.
Assume Benjamin is a knave. No contradiction is found in their false claim that Lucas is a knave.
Lucas cannot be a knight, because this would contradict the claim of their own that Aiden is not a knight.
Lucas cannot be a knave, because this would contradict the false claim of Benjamin that Lucas is a knave.
We have exhausted all possibilities for Lucas and Benjamin, so let us go back and reconsider Aiden.
Assume Aiden is a knave. No contradiction is found in their false claim that Benjamin is a knight if and only if Grace is a knight.
Assume Benjamin is a knight. No contradiction is found in their claim that Lucas is a knave.
Lucas cannot be a knight, because this would contradict the claim of Benjamin that Lucas is a knave.
Lucas cannot be a knave, because this would contradict the false claim of their own that Aiden is not a knight.
We have exhausted all possibilities for Lucas, so let us go back and reconsider Benjamin.
Assume Benjamin is a knave. No contradiction is found in their false claim that Lucas is a knave.
Assume Lucas is a knight. No contradiction is found in their claim that Aiden is not a knight.
Assume Grace is a knight. No contradiction is found in their claim that Victoria is a knave.
Victoria cannot be a knight, because this would contradict the claim of their own that Aiden is a knight.
Assume Victoria is a knave. No contradiction is found in their false claim that Aiden is a knight.
Therefore, (1) Aiden is a knave
(2) Victoria is a knave
(3) Lucas is a knight
(4) Benjamin is a knave
(5) Grace is a knight </think>
<answer> (1) Aiden is a knave
(2) Victoria is a knave
(3) Lucas is a knight
(4) Benjamin is a knave
(5) Grace is a knight </answer>
Aiden Victoria Lucas Benjamin Grace
(1) Aiden is a knave
(2) Victoria is a knave
(3) Lucas is a knight
(4) Benjamin is a knave
(5) Grace is a knight
(('<=>', ('telling-truth', 3), ('telling-truth', 4)), ('telling-truth', 0), ('not', ('telling-truth', 0)), ('lying', 2), ('lying', 1))
rule