
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 the sequences ${a_n}$ and ${b_n}$ that satisfy ${a_1} = {b_1} = 1$, ${a_2} = 3$, $S_n$ is the sum of the first $n$ terms of the sequence ${a_n}$, and $S_{n+1} + S_{n-1} = 2(S_n + 1) (n \geqslant 2, n \in \mathbb{N})$, also ${b_1} + 2{b_2} + {2^2}{b_3} + \cdots + {2^{n - 2}}{b_{n - 1}} + {2^{n - 1}}{b_n} = {a_n}$ holds for any $n \in \mathbb{N}$.

(1) Find the general terms of the sequences ${a_n}$ and ${b_n}$.
(2) 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}
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