
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
Assume $y_1, y_2, \ldots, y_n$ are nonnegative real numbers such that the sum \(y_1 + y_2 + \ldots + y_n = 2\) and 
\[
y_1^2 + y_2^2 + \ldots + y_n^2 \le \frac{2}{50}.
\]
Determine the smallest possible value of $n$.

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