
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a castle, there are 16 identical square rooms arranged in a $4 \times 4$ square. Sixteen people—some knights and some liars—each occupy one room. Liars always lie, and knights always tell the truth. Each of these 16 people said, "At least one of my neighboring rooms houses a liar." Rooms are considered neighbors if they share a common wall. What is the maximum number of knights that could be among these 16 people?

Present the answer in LaTex format: \boxed{Your answer}
user