
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 $\{a_n\}$ with the first term $a_1=1$ and a common difference $d>0$, and a geometric sequence $\{b_n\}$, satisfying $b_2=a_2$, $b_3=a_5$, $b_4=a_{14}$.  
(1) Find the general term of the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Let the sequence $\{c_n\}$ satisfy $c_n=2a_n-18$, find the minimum value of the sum of the first $n$ terms $S_n$ of the sequence $\{c_n\}$, and the value of $n$ at this minimum.

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