
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
Given the sequence $\{a_n\}$ where $a_1 = 2$, $a_n - a_{n-1} - 2n = 0$ ($n \geq 2$, $n \in \mathbb{N}^*$).  
(1) Write out the values of $a_2$, $a_3$ (results only), and derive the general term formula for the sequence $\{a_n\}$.  
(2) Let $b_n = \frac{1}{a_{n+1}} + \frac{1}{a_{n+2}} + \dots + \frac{1}{a_{2n}}$. If the inequality $t^2 - 2t + \frac{1}{6} > b_n$ always holds true for any positive integer $n$, find the range of values for the real number $t$.

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