
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 ${{a}_{n}}$ that satisfies ${{a}_{1}}=3,\sqrt{{{a}_{n+1}}+1}-\sqrt{{{a}_{n}}+1}=1,n\in {{N}^{*}}$, find the general term formula for the sequence. Additionally, if ${{b}_{n}}={{\log }_{2}}\frac{{{n}^{2}}+n}{{{a}_{n}}}$, and the sum of the first $n$ terms of the sequence ${{b}_{n}}$ is denoted as ${{S}_{n}}$, find the smallest natural number $n$ such that ${{S}_{n}}<-4$.

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