
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 positive real numbers $x$ and $y$ satisfy $\lg x + \lg y = 8$, and $S_{n} = \lg x^{n} + \lg (x^{n-1}y) + \lg (x^{n-2}y^{2}) + \ldots + \lg (xy^{n-1}) + \lg y^{n}$, where $n \in \mathbb{N}^{*}$, then $S_{n} =$ _________.

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