
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
The sequences $\{a_n\}$ and $\{b_n\}$ satisfy: $a_n + b_n = 2n - 1$, where $n \in \mathbb{N}^*$.  
(1) If the sum of the first $n$ terms of $\{a_n\}$, denoted as $S_n$, equals $2n^2 - n$, find the general formulas for $\{a_n\}$ and $\{b_n\}$.  
(2) If $a_n = k \cdot 2^{n-1}$, where $n \in \mathbb{N}^*$, and the sequence $\{b_n\}$ is monotonically decreasing, find the range of the real number $k$.

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