
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 two positive sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfying $\frac{1}{{{a_n}-{b_n}}}={b_n}$ and $\frac{1}{{{a_n}}}=\frac{{{b_n}}}{{{n^2}+1}$. Find:<br/>
$(1)$ The general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;<br/>
$(2)$ Let $\left[x\right]$ represent the greatest integer not exceeding $x$. Find the sum $S_{n}$ of the first $n$ terms of the sequence $\{[a_{n}+a_{n+1}]\cdot {2}^{{b}_{n}}$.

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