
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 term 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 $n$ such that $y_n > \frac{55}{9}$.

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