
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 ${S_n}$ be the sum of the first $n$ terms of the sequence ${a_n}$, and ${a_1 = 1, a_{n+1} = -S_n S_{n+1}}$. Find the value of $n$ that maximizes $\frac{nS_n^2}{1 + 10S_n^2}$.

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