
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 sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=4$, and $a_{n+2}+2a_n=3a_{n+1}$, where $n \in \mathbb{N}^*$. 

(1) Find the general formula for the sequence $\{a_n\}$.

(2) Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Find the smallest integer $n$ such that $S_n > 21-2n$.

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