
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 $\frac{1^2}{1 \times 3}$, $\frac{2^2}{3 \times 5}$, $\frac{3^2}{5 \times 7}$, ..., $\frac{n^2}{(2n-1) \times (2n+1)}$, ..., let $S_n$ denote the sum of the first $n$ terms. It is computed that $S_1 = \frac{1}{3}$, $S_2 = \frac{3}{5}$, $S_3 = \frac{6}{7}$, $S_4 = \frac{10}{9}$. Observe these results and deduce the formula for $S_n =$ \_\_\_\_\_\_.

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