
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 two sequences of positive numbers \(\{a_{n}\}\) and \(\{b_{n}\}\) that satisfy:
(a) \(a_{0} = 1 \geqslant a_{1}\), and \(a_{n}(b_{n-1} + b_{n+1}) = a_{n-1}b_{n-1} + a_{n+1}b_{n+1}\) for \(n \geqslant 1\);
(b) \(\sum_{i=0}^{n} b_{i} \leqslant n^{\frac{3}{2}}\) for \(n \geqslant 1\),

find the general term for the sequence \(\{a_{n}\}\).

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