
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 that the arithmetic sequence $\{a_n\}$ satisfies $S_9=117$, $a_7=19$, and the sequence $\{b_n\}$ satisfies $\sum\limits_{i=1}^{n}2^{i-1}b_{i}=n$.
(I) Find the general term formulas for the sequences $\{a_n\}$ and $\{b_n\}$;
(II) Find the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n a_{n+1}}+b_n\}$.

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