
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 \( a_{1}, a_{2}, a_{3}, \cdots \) that satisfy the following two conditions:
(1) \( a_{n} \leqslant n \sqrt{n} \) for \( n=1,2,3, \cdots \)
(2) For any different \( m \) and \( n \), \( m-n \) divides \( a_{m} - a_{n} \).

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