
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
People call the number $\frac{{\sqrt{5}-1}}{2}\approx 0.618$ the golden ratio. The "$0.618$ method" in the famous mathematician Hua Luogeng's optimal selection method applies the golden ratio. Let $a=\frac{{\sqrt{5}-1}}{2}$, $b=\frac{{\sqrt{5}+1}}{2}$. Define $S_{1}=\frac{1}{{1+a}}+\frac{1}{{1+b}}$, $S_{2}=\frac{2}{{1+{a^2}}}+\frac{2}{{1+{b^2}}}$, $\ldots$, $S_{100}=\frac{{100}}{{1+{a^{100}}}}+\frac{{100}}{{1+{b^{100}}}}$. Find the value of $S_{1}+S_{2}+\ldots +S_{100}$.

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