
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 positive numbers \(a_{1}, a_{2}, \ldots, a_{n}\) and a real number \(q\) which satisfies the double inequality \(0 < q < 1\), find \(n\) real numbers \(b_{1}, b_{2}, \ldots, b_{n}\) that simultaneously satisfy the following conditions:
a) \(a_{k} < b_{k} \quad (k=1,2, \ldots, n)\)
b) \(q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q} \quad (k=1,2, \ldots, n-1)\)
c) \(b_{1} + b_{2} + \ldots + b_{n} < \frac{1+q}{1-q} \left(a_{1} + a_{2} + \ldots + a_{n}\right)\)

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