
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
There are 33 knights, liars, and dreamers living on an island. Each resident was asked in turn, "How many knights are there among you?" Ten different answers were received, each of which was given by more than one resident. Knights always tell the truth, liars always give an incorrect number that hasn't been mentioned yet, and dreamers always give a number that is one greater than the previous answer. Was the number 40 necessarily mentioned?

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