
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 values of \( n \), \( n \in \mathbb{N} \), for which the sum of the first \( n \) terms of the sequence \( a_k = 3k^2 - 3k + 1 \), \( k \in \mathbb{N} \), is equal to the sum of the first \( n \) terms of the sequence \( b_k = 2k + 89 \), \( k \in \mathbb{N} \). (12 points)

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