
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
Let \( a_n = 2^n \), where \( n \in \mathbb{N}^{+} \). The sequence \(\{b_n\}\) satisfies the relation \( b_1 a_n + b_2 a_{n-1} + \cdots + b_n a_1 = 2^n - \frac{n}{2} - 1 \). Find the sum of the first \( n \) terms of the sequence \(\{a_n \cdot b_n\}\).

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