
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 $\{a_n\}$ satisfying $a_1=3$ and $\sqrt{a_{n+1}+1} - \sqrt{a_{n}+1} = 1$ for $n\in \mathbb{N}^*$:
1. Find the general term formula for the sequence $\{a_n\}$.
2. Let $b_n = \log_2 \frac{n^2+n}{a_n}$ and the sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$. Find the smallest natural number $n$ such that $S_n < -4$.

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