
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 $S_n$, and $S_n = 2a_n - \frac{1}{2}$ $(n \in \mathbb{N}^*)$, and the sequence $\{b_n\}$ satisfies $b_1 = 1$, and the point $P(b_n, b_{n+1})$ is on the line $x - y + 2 = 0$.

1. Find $a_n$ 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 as $T_n$;
3. If $\lambda > 0$, find the range of $k$ such that $a_{2n}(2\lambda^2 - k\lambda + 2) > b_n$ holds for all positive integers $n$.

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