
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
Given: $\alpha $, $\beta \left(\alpha\ \ \gt \beta \right)$ are the two real roots of the quadratic equation $x^{2}-x-1=0$, $s_{1}=\alpha +\beta $, $s_{2}=\alpha ^{2}+\beta ^{2}$, ..., $s_{n}=\alpha ^{n}+\beta ^{n}$.  
$(1)$ Write down the values of $s_{1}$ and $s_{2}$ directly: $s_{1}=$______, $s_{2}=$______;  
$(2)$ After calculation, it is found that $s_{3}=4$, $s_{4}=7$, $s_{5}=11$. When $n\geqslant 3$, please make a conjecture about the quantitative relationship between $S_{n}$, $S_{n-1}$, and $S_{n-2}$, and provide a proof.

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