
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 $\{b_n\}$ and a sequence $\{a_n\}$ where all terms are positive, and $a_1=b_1=1$, $b_2+b_4=10$, satisfying $a_n^2-2a_na_{n+1}+a_n-2a_{n+1}=0$  
(1) Find the general formula for $\{a_n\}$ and $\{b_n\}$.  
(2) Let $c_n= \frac{1}{a_n}+b_n$, calculate the sum of the first $n$ terms of the sequence $\{c_n\}$.

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