
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
Let \( a_{1} = 3 \), and for \( n \geq 1 \), let \( a_{n+1} = (n+1) a_{n} - n \). Find the smallest \( m \geq 2005 \) such that \( a_{m+1} - 1 \mid a_{m}^{2} - 1 \).

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