
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 sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}=\frac{n}{n+1}$ for $n\in \mathbb{N}_{+}$, and the sequence $\{b_{n}\}$ satisfies $b_{1}=1$ and $b_{n+1}=\frac{b_{n}}{b_{n}+2}$ for $n\in \mathbb{N}_{+}$. <br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$; <br/>$(2)$ Find the general formula for the sequence $\{b_{n}\}$; <br/>$(3)$ For $n\in \mathbb{N}_{+}$, compare the sizes of $b_{n+1}$ and $a_{n}$.

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