
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
The sequence \(\left\{a_{n}\right\}\) satisfies \(a_{1}=1, \sqrt{\frac{1}{a_{n}^{2}}+4}=\frac{1}{a_{n+1}}\). Find the smallest positive integer \(t\) such that the inequality \(S_{2n+1}-S_{n} \leqslant \frac{t}{30}\) holds for any \(n \in \mathbf{N}^{*}\), where \(S_{n}=\sum_{i=1}^{n} a_{i}^{2}\).

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