
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 that for any $n \in \mathbb{N}^{*}$, the vector $\overrightarrow{d_{n}} = \left(a_{n+1} - \frac{1}{4}a_{n}, \frac{a_{n+1}^{2}}{a_{n}}\right)$ is always a direction vector of the line $y=x$. Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$. If $a_{1} \neq 1$, then $\lim_{n \to \infty} S_{n} = \_\_\_\_\_\_$.

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