Find the largest positive real number $\lambda$ such that for any nonnegative real numbers $x,$ $y,$ and $z$ such that $x^2 + y^2 + z^2 = 1,$ the inequality
\[\lambda xy + yz \le \frac{\sqrt{5}}{2}\]holds.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the largest positive real number $\lambda$ such that for any nonnegative real numbers $x,$ $y,$ and $z$ such that $x^2 + y^2 + z^2 = 1,$ the inequality
\[\lambda xy + yz \le \frac{\sqrt{5}}{2}\]holds.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2
Find the largest positive real number $\lambda$ such that for any nonnegative real numbers $x,$ $y,$ and $z$ such that $x^2 + y^2 + z^2 = 1,$ the inequality
\[\lambda xy + yz \le \frac{\sqrt{5}}{2}\]holds.
train