
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 $S_{n}$, $S_{n}=2a_{n}-2(n\in N^{*})$, and the sequence $\{b_{n}\}$ satisfies $b_{1}=1$, the point $P(b_{n}, b_{n+1})$ lies on the line $x-y+2=0$.
- $(1)$ Find the general terms $a_{n}$ and $b_{n}$ for the sequences $\{a_{n}\}$ and $\{b_{n}\}$.
- $(2)$ Let $c_{n}=a_{n}\cdot b_{n}$. Find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $T_{n}$.
- $(3)$ If $\lambda\ \ \gt 0$, find the range of $k$ for which $2\lambda ^{2}-k\lambda +2 \gt \frac{{b_n}}{{a_{2n}}}$ holds for all positive integers $n$.

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