
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 the royal dining hall, there are three tables with three identical pies. For lunch, the king invites six princes to his table. At the second table, there can be 12 to 18 courtiers, and at the third table, there can be 10 to 20 knights. Each pie is divided into equal pieces according to the number of people seated at the table. There is a rule at the court: the amount of food a knight receives combined with the amount a courtier receives equals the amount received by the king. Determine the maximum possible number of knights the king can invite for lunch that day. How many courtiers will be seated at their table in this case?

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