
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
The sequence $\{a_n\}$ is an arithmetic sequence with the first term being $1$ and the common difference being $2$. $S_n$ is the sum of the first $n$ terms. Then, $\lim\limits_{n \to \infty} \frac{S_n}{a_n^2} =$ \_\_\_\_\_\_.

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