
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 $S\_n$, where $S\_n = 2a\_n - \frac{1}{2}\ (n \in \mathbb{N}^*)$, and the sequence $\{b\_n\}$ satisfies $b\_1 = 1$, with the point $P(b\_n, b\_n+1)$ 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, $T\_n$, for the sequence $\{c\_n\}$.

(3) If $\lambda > 0$, find the range of $k$ such that $2\lambda^2 - k\lambda + 2 > \frac{b\_n}{a\_{2n}}$ holds for all positive integers $n$.

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