
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 a sequence $1$, $1$, $3$, $1$, $3$, $5$, $1$, $3$, $5$, $7$, $1$, $3$, $5$, $7$, $9$, $\ldots$, where the first term is $1$, the next two terms are $1$, $3$, and the next three terms are $1$, $3$, $5$, and so on. Let $S_{n}$ denote the sum of the first $n$ terms of this sequence. Find the smallest positive integer value of $n$ such that $S_{n} > 400$.

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