
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
A three-digit number, with distinct digits, has its middle digit equal to the arithmetic mean of the outer digits. Write the last digit in front of the first two, and form the geometric mean of the newly formed number and the original number. This results in the number obtained by moving the first digit of the original number behind the other two. Determine the digits. Are there such sets of three-digit numbers in other numeral systems as well?

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