
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 math lesson, each gnome needs to find a three-digit number without zero digits, divisible by 3, such that when 297 is added to it, the resulting number consists of the same digits but in reverse order. What is the minimum number of gnomes that must be in the lesson so that among the numbers they find, there are always at least two identical ones?

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