
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 an arithmetic sequence $\{a_n\}$ and an increasing geometric sequence $\{b_n\}$ satisfying: $a_1=1$, $b_1=3$, and $b_3=2a_5+3a_2$, $b_2=a_4+2$  
$(1)$ Find the general formulas for sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let $S_n$ denote the sum of the first $n$ terms of sequence $\{a_n\}$. If for any $n\in\mathbb{N}^*$, $kb_n \geqslant S_n$ always holds, find the range of the real number $k$.

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