
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 an arithmetic sequence $\{a_n\}$ and a geometric sequence $\{b_n\}$ satisfying: $a_1 = b_1 = 1$, $a_2 = b_2$, and $2a_3 - b_3 = 1$.
(Ⅰ) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$;
(Ⅱ) Denoting $c_n = a_n b_n$, calculate the sum $S_n$ of the first $n$ terms of the sequence $\{c_n\}$.

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