
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
The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, $a_1=1$, $a_{n+1}=2S_n+1$, and the arithmetic sequence $\{b_n\}$ satisfies $b_3=3$, $b_5=9$,

(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$ respectively;  
(2) If for any $n\in N^*$, $(S_n+ \frac{1}{2})\cdot k \geqslant b_n$ always holds, find the range of the real number $k$.

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