
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 $\{a_{n}\}$ where $a_{3}=S_{2}+1$ and $S_{3}=a_{4}+2$, where $S_{n}$ is the sum of the first $n$ terms of $\{a_{n}\}$, and an increasing geometric sequence $\{b_{n}\}$ with $b_{1}=1$ and $b_{1}$, $b_{2}$, $b_{3}-4$ form an arithmetic sequence.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\cdot b_{n}\}$, find $T_{n}$;<br/>$(3)$ Let $C_{n}=\frac{{({a_n}+4)}}{{({S_n}+n)•{b_{n+1}}}}$, $A_{n}$ be the sum of the first $n$ terms of $\{C_{n}\}$, where $A_{n}\geqslant \frac{λ}{{n+1}}$ always holds, find the maximum value of the real number $\lambda$.

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