
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 studying square roots, Xiaoming found that some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}=(1+\sqrt{2})^2$. Xiaoming, who is good at thinking, conducted the following exploration: Let $a+b\sqrt{2}=(m+n\sqrt{2})^2$ (where $a$, $b$, $m$, $n$ are all integers), then we have $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$, therefore $a=m^{2}+2n^{2}$, $b=2mn$. In this way, Xiaoming found a method to square expressions similar to $a+b\sqrt{2}$. Please follow Xiaoming's method to explore and solve the following problems:<br/>$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=(m+n\sqrt{3})^2$, express $a$, $b$ in terms of $m$, $n$, and find $a=$______, $b=$______;<br/>$(2)$ If $a+4\sqrt{3}=(m+n\sqrt{3})^2$, and $a$, $m$, $n$ are all positive integers, find the value of $a$.<br/>$(3)$ Simplify $\sqrt{6+2\sqrt{5}}$.

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