
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
After reading the following solution, answer the question: Xiaofang found in the simplification of $\sqrt{7+4\sqrt{3}}$ that first, $\sqrt{7+4\sqrt{3}}$ can be simplified to $\sqrt{7+2\sqrt{12}}$. Since $4+3=7$ and $4\times 3=12$, that is, ${(\sqrt{4})^2}+{(\sqrt{3})^2}=7$, $\sqrt{4}×\sqrt{3}=\sqrt{12}$, so $\sqrt{7+4\sqrt{3}}=\sqrt{7+2\sqrt{12}}=\sqrt{{{(\sqrt{4})}^2}+2\sqrt{4×3}+{{(\sqrt{3})}^2}}=\sqrt{{{(\sqrt{4}+\sqrt{3})}^2}}=2+\sqrt{3}$. The question is:<br/>$(1)$ Fill in the blanks: $\sqrt{4+2\sqrt{3}}=$______, $\sqrt{5-2\sqrt{6}}=$______; <br/>$(2)$ Further research reveals that the simplification of expressions in the form of $\sqrt{m±2\sqrt{n}}$ can be done by finding two positive numbers $a$ and $b\left(a \gt b\right)$ such that $a+b=m$, $ab=n$, that is, ${(\sqrt{a})^2}+{(\sqrt{b})^2}=m$, $\sqrt{a}×\sqrt{b}=\sqrt{n}$, then we have: $\sqrt{m±2\sqrt{n}}=\_\_\_\_\_\_.$<br/>$(3)$ Simplify: $\sqrt{4-\sqrt{15}$ (Please write down the simplification process).

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