
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 $S_{n}$, when $n\in N^{*}$, $S_{n}+2=2a_{n}$; in the sequence $\{b_{n}\}$, $b_{1}=1$. The line $x-y+1=0$ passes through the point $P(b_{n}$,$b_{n+1})$.  
$(1)$ Find the general formulas for the sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let $c_{n}=a_{n}b_{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$, denoted as $T_{n}$, and determine the largest integer $n$ such that $T_{n} \lt 2022$.

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