
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 \( n \geq 4 \), and \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) be two sets of real numbers, satisfying \(\sum_{j=1}^{n} \alpha_{j}^{2}<1\) and \(\sum_{j=1}^{n} \beta_{j}^{2}<1\). Define:
\[ A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, \]
\[ B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, \]
\[ W=\frac{1}{2}\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2}. \]

Find all real numbers \(\lambda\) such that the equation
\[ x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0 \]
has only real roots.

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