
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
Find all sequences of natural numbers \( (a_n) \) such that the following two conditions are satisfied:

a) For any \( n \), \( a_{n} \leq n \sqrt{n} \);

b) For any distinct \( m \) and \( n \), the difference \( a_{m} - a_{n} \) is divisible by \( m - n \).

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