
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
Reading material: After studying quadratic radicals, Xiaoming found that some expressions containing radicals can be written as the square of another expression, such as $3+2\sqrt{2}=(1+\sqrt{2})^{2}$. In this way, $\sqrt{3+2\sqrt{2}}$ can be simplified as $\sqrt{(1+\sqrt{2})^{2}}=1+\sqrt{2}$. The thoughtful Xiaoming conducted the following exploration: For $a+2\sqrt{b}$, if two numbers $m$ and $n$ can be found such that $m^{2}+n^{2}=a$ and $mn=\sqrt{b}$, then $a+2\sqrt{b}$ can be transformed into $m^{2}+n^{2}+2mn$, that is, $\left(m+n\right)^{2}$, thus making $\sqrt{a+2\sqrt{b}}=\sqrt{(m+n)^{2}}=m+n$ (where $a$, $b$, $m$, $n$ are all positive integers). For example: $\because 4+2\sqrt{3}=1+3+2\sqrt{3}=(\sqrt{1})^{2}+(\sqrt{3})^{2}+2\sqrt{3}=(1+\sqrt{3})^{2}$, $\therefore \sqrt{4+2\sqrt{3}}=\sqrt{(1+\sqrt{3})^{2}}=1+\sqrt{3}$. Please explore and solve the following problems following Xiaoming's method: $(1)$ Simplify $\sqrt{6+2\sqrt{5}$; $(2)$ Simplify $\sqrt{5-2\sqrt{6}$; $(3)$ If $\sqrt{{a}^{2}+4\sqrt{5}}=2+\sqrt{5}$, find the value of $a$.

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