
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 a sequence $\{a_n\}$ that satisfies $a_1=1$, $a_2=4$, and $a_{n+2}+2a_n=3a_{n+1}$ for $n \in \mathbb{N}^+$, and let $b_n = a_{n+1} - a_n$,

1. Find the general formula for the sequences $\{b_n\}$ and $\{a_n\}$;
2. Let the sum of the first $n$ terms of $\{a_n\}$ be $S_n$, find the smallest integer $n$ such that $S_n > 21 - 2n$.

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