
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 \(\lambda > 0\). Find the largest constant \(c = c(\lambda)\) such that for all non-negative real numbers \(x\) and \(y\), the inequality \(x^{2} + y^{2} + \lambda x y \geqslant c(x + y)^{2}\) holds.

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