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 7 inhabitants: Jackson, Grace, Logan, Jack, Emily, Emma, and William. Jackson asserted: "Emma is a knight if and only if William is a knave". Grace remarked, "If Jackson is a knight then Grace is a knight". Logan said, "Grace is not a knave." Jack noted, "Emma is a knight or William is a knave". Emily said that Jackson is a knight or Emma is a knave. Emma was heard saying, "If Grace is a knight then William is a knight". William said that Jack is a knave or Emma 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 Jackson is a knight.  No contradiction is found in their claim that Emma is a knight if and only if William is a knave.
 Assume Emma is a knight.  No contradiction is found in their claim that If Grace is a knight then William is a knight.
 Assume Grace is a knight.  No contradiction is found in their claim that If Jackson is a knight then Grace is a knight.
 William cannot be a knight, because this would contradict the claim of Jackson that Emma is a knight if and only if William is a knave.
 William cannot be a knave, because this would contradict the claim of Emma that If Grace is a knight then William is a knight.
 We have exhausted all possibilities for William, so let us go back and reconsider Grace.
 Assume Grace is a knave.  No contradiction is found in their false claim that If Jackson is a knight then Grace is a knight.
 William cannot be a knight, because this would contradict the claim of Jackson that Emma is a knight if and only if William is a knave.
 Assume William is a knave.  No contradiction is found in their false claim that Jack is a knave or Emma is a knave.
 Assume Jack is a knight.  No contradiction is found in their claim that Emma is a knight or William is a knave.
 Logan cannot be a knight, because this would contradict the claim of their own that Grace is not a knave.
 Assume Logan is a knave.  No contradiction is found in their false claim that Grace is not a knave.
 Assume Emily is a knight.  No contradiction is found in their claim that Jackson is a knight or Emma is a knave.
 Therefore, (1) Jackson is a knight
 (2) Grace is a knave
 (3) Logan is a knave
 (4) Jack is a knight
 (5) Emily is a knight
 (6) Emma is a knight
 (7) William is a knave
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 7 inhabitants: Jackson, Grace, Logan, Jack, Emily, Emma, and William. Jackson asserted: "Emma is a knight if and only if William is a knave". Grace remarked, "If Jackson is a knight then Grace is a knight". Logan said, "Grace is not a knave." Jack noted, "Emma is a knight or William is a knave". Emily said that Jackson is a knight or Emma is a knave. Emma was heard saying, "If Grace is a knight then William is a knight". William said that Jack is a knave or Emma 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 Jackson is a knight. No contradiction is found in their claim that Emma is a knight if and only if William is a knave.
Assume Emma is a knight. No contradiction is found in their claim that If Grace is a knight then William is a knight.
Assume Grace is a knight. No contradiction is found in their claim that If Jackson is a knight then Grace is a knight.
William cannot be a knight, because this would contradict the claim of Jackson that Emma is a knight if and only if William is a knave.
William cannot be a knave, because this would contradict the claim of Emma that If Grace is a knight then William is a knight.
We have exhausted all possibilities for William, so let us go back and reconsider Grace.
Assume Grace is a knave. No contradiction is found in their false claim that If Jackson is a knight then Grace is a knight.
William cannot be a knight, because this would contradict the claim of Jackson that Emma is a knight if and only if William is a knave.
Assume William is a knave. No contradiction is found in their false claim that Jack is a knave or Emma is a knave.
Assume Jack is a knight. No contradiction is found in their claim that Emma is a knight or William is a knave.
Logan cannot be a knight, because this would contradict the claim of their own that Grace is not a knave.
Assume Logan is a knave. No contradiction is found in their false claim that Grace is not a knave.
Assume Emily is a knight. No contradiction is found in their claim that Jackson is a knight or Emma is a knave.
Therefore, (1) Jackson is a knight
(2) Grace is a knave
(3) Logan is a knave
(4) Jack is a knight
(5) Emily is a knight
(6) Emma is a knight
(7) William is a knave </think>
<answer> (1) Jackson is a knight
(2) Grace is a knave
(3) Logan is a knave
(4) Jack is a knight
(5) Emily is a knight
(6) Emma is a knight
(7) William is a knave </answer>
Jackson Grace Logan Jack Emily Emma William
(1) Jackson is a knight
(2) Grace is a knave
(3) Logan is a knave
(4) Jack is a knight
(5) Emily is a knight
(6) Emma is a knight
(7) William is a knave
(('<=>', ('telling-truth', 5), ('lying', 6)), ('->', ('telling-truth', 0), ('telling-truth', 1)), ('not', ('lying', 1)), ('or', ('telling-truth', 5), ('lying', 6)), ('or', ('telling-truth', 0), ('lying', 5)), ('->', ('telling-truth', 1), ('telling-truth', 6)), ('or', ('lying', 3), ('lying', 5)))
rule