
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
Determine the smallest positive real number $k$ such that
\[
y_1^2 + y_2^2 + y_3^2 \geq kA^2
\]
whenever $y_1, y_2, y_3$ are real numbers such that $y_1 + y_2 + y_3 = 0$ and $A = \max(|y_1|, |y_2|, |y_3|)$.

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