
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
Let the sequence $\{x_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and it satisfies $4x_n-S_n-3=0$ for $n\in\mathbb{N}^*$;  
(1) Find the general formula for the sequence $\{x_n\}$;  
(2) If the sequence $\{y_n\}$ satisfies $y_{n+1}-y_n=x_n$ for $n\in\mathbb{N}^*$, and $y_1=2$, find the smallest positive integer $n$ that satisfies the inequality $y_n > \frac{55}{9}$.

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