
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 $a \geq 2$ be a real number, and let $x_1$ and $x_2$ be the roots of the equation $x^{2} - a x + 1 = 0$. Define $S_{n} = x_{1}^{n} + x_{2}^{n}$ for $n = 1, 2, \cdots$.

(1) Determine the monotonicity of the sequence $\left\{\frac{S_{n}}{S_{n+1}}\right\}$ and provide a proof.

(2) Find all real numbers $a$ such that for any positive integer $n$, certain conditions are satisfied.

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