
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
The math teacher asked the students to create a math problem based on the content related to quadratic radicals. Below are two questions selected by the math teacher and one question she created herself. Read them first and then answer the questions.

$(1)$ Xiaoying's question. Observe the following equations:  
$\frac{2}{{\sqrt{3}+1}}=\frac{{2(\sqrt{3}-1)}}{{(\sqrt{3}+1)(\sqrt{3}-1)}}=\frac{{2(\sqrt{3}-1)}}{{{{(\sqrt{3})}^2}-{1^2}}}=\frac{{2(\sqrt{3}-1)}}{{3-1}}=\sqrt{3}-1$;  
$\frac{2}{{\sqrt{5}+\sqrt{3}}}=\frac{{2(\sqrt{5}-\sqrt{3})}}{{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}}=\frac{{2(\sqrt{5}-\sqrt{3})}}{{{{(\sqrt{5})}^2}-{{(\sqrt{3})}^2}}}=\frac{{2(\sqrt{5}-\sqrt{3})}}{{5-3}}=\sqrt{5}-\sqrt{3}$;  
Write down the result of the following expression directly: $\frac{2}{{\sqrt{7}+\sqrt{5}}}=$______;

$(2)$ Xiaoming's question. According to the multiplication of quadratic radicals:  
${(\sqrt{3}+1)^2}=4+2\sqrt{3}$, ${(\sqrt{5}+\sqrt{3})^2}=8+2\sqrt{15}$, ${(\sqrt{a}+\sqrt{b})^2}=a+b+2\sqrt{ab}(a≥0,b≥0)$  
According to the definition of square roots, we have $\sqrt{4+2\sqrt{3}}=\sqrt{3}+1$, $\sqrt{8+2\sqrt{15}}=\sqrt{5}+\sqrt{3}$, $\sqrt{a+b+2\sqrt{ab}}=\sqrt{a}+\sqrt{b}(a≥0,b≥0)$;  
Write down the result of the following expression directly: $\sqrt{7+4\sqrt{3}}=$______;

$(3)$ The math teacher's question. Based on your findings, complete the following calculation: $({\frac{2}{{\sqrt{3}+1}}+\frac{2}{{\sqrt{5}+\sqrt{3}}}+\frac{2}{{\sqrt{7}+\sqrt{5}}}+\frac{2}{{\sqrt{9}+\sqrt{7}}}+\frac{2}{{\sqrt{11}+\sqrt{9}}}})⋅\sqrt{12+2\sqrt{11}}$.

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