
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 sequence $\{a_n\}$ is an arithmetic sequence, and the sequence $\{b_n\}$ satisfies $b_n = a_n + n$. If $b_2$, $b_5$, and $b_{11}$ form a geometric sequence, and $b_3 = a_6$, find:
(1) $a_n$ and $b_n$;
(2) The sum of the first $n$ terms, $S_n$, of the sequence $\left\{\frac{1}{a_nb_n}\right\}$.

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