
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  $a_1, \ldots, a_n$  be  $n$  positive numbers and  $0 < q < 1.$  Determine  $n$  positive numbers  $b_1, \ldots, b_n$  so that:

*a.)*  $ a_{k} < b_{k}$  for all  $k = 1, \ldots, n,$  
*b.)*  $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$  for all  $k = 1, \ldots, n-1,$ *c.)*  $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$ 

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