
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 $\{a_n\}$ be a positive geometric sequence, given that $a_2=2$, and $a_3a_4a_5=2^9$.  
(1) Find the first term $a_1$ and the common ratio $q$.  
(2) If the sequence $\{b_n\}$ satisfies $b_n= \frac{1}{n}[ \lg a_1 + \lg a_2 + \ldots + \lg a_{n-1} + \lg (ka_n) ]$, is there a positive number $k$ such that $\{b_n\}$ forms an arithmetic sequence? If so, find the value of $k$. If not, explain why.

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