
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

Let \( a_{1}, a_{2}, \ldots, a_{n} \) be positive numbers and \( q \) a given real number, \( 0 < q < 1 \). Find \( n \) real numbers \( b_{1}, b_{2}, \ldots, b_{n} \) that satisfy:

1. \( a_{k} < b_{k} \) for all \( k = 1, 2, \ldots, n \)
2. \( q < \frac{b_{k}+1}{b_{k}} < \frac{1}{q} \) for all \( k = 1, 2, \ldots, n-1 \)
3. \( b_{1} + b_{2} + \cdots + b_{n} < \frac{1 + q}{1 - q} \left( a_{1} + a_{2} + \cdots + a_{n} \right) \)

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