
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 $\{a_n\}$ is an increasing arithmetic sequence and satisfies $a_2 a_4 = 21$, $a_1 + a_5 = 10$.  
(1) Find the general formula for the n-th term of the sequence $\{a_n\}$.  
(2) If the sequence $\{c_n\}$ satisfies that the sum of its first n terms $C_n = a_n + 1$, and the sequence $\{b_n\}$ is defined by $b_n = 2^n c_n$ for $n \in \mathbb{N}^*$, find the sum of the first n terms of the sequence $\{b_n\}$.

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