
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 square roots, Xiaoming found that some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}=(1+\sqrt{2})^{2}$. With his good thinking skills, Xiaoming explored the following: if $a+b\sqrt{2}=(m+n\sqrt{2})^{2}=m^{2}+2n^{2}+2mn\sqrt{2}$ (where $a$, $b$, $m$, $n$ are all integers), then $a=m^{2}+2n^{2}$ and $b=2mn$. In this way, Xiaoming found a method to transform expressions like $a+b\sqrt{2}$ into squared forms. Please follow Xiaoming's method to explore and solve the following problems:  
[Problem Solving]  
$(1)$ If $a+b\sqrt{5}=(m+n\sqrt{5})^{2}$, when $a$, $b$, $m$, $n$ are all integers, then $a=\_\_\_\_\_\_$, $b=\_\_\_\_\_\_$ (expressed using expressions containing $m$ and $n).  
$(2)$ If $x+4\sqrt{3}=(m+n\sqrt{3})^{2}$, and $x$, $m$, $n$ are all positive integers, find the values of $x$, $m$, $n$ respectively.  
[Extension]  
$(3)$ Simplify $\sqrt{5+2\sqrt{6}}=\_\_\_\_\_\_$.

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