
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 a sequence \(\left\{a_{n}\right\}\left(a_{n} \geqslant 0\right)\) that satisfies \(a_{1}=0\), \(a_{2}=1\), \(a_{3}=9\), and the relation
\[ S_{n}^{2} S_{n-2}=10 S_{n-1}^{3} \quad (n>3), \]
where \(S_{n}\) represents the sum of the first \(n\) terms of the sequence \(\left\{a_{n}\right\}\), find the expression for \(a_{n} \) for \(n \geqslant 3\).

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