
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
For certain natural numbers \( n > m \), the number \( n \) can be represented both as the sum of 2021 terms, each of which is an integer non-negative power of \( m \), and as the sum of 2021 terms, each of which is an integer non-negative power of \( m+1 \). What is the largest possible value of \( m \) for which this can happen (for some \( n > m \))?

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