
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 roots, Xiao Li found that some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}=(1+\sqrt{2})^2$. Xiao Li, 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 $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$. Therefore, $a=m^{2}+2n^{2}$, $b=2mn$. In this way, Xiao Li found a method to transform expressions like $a+b\sqrt{2}$ into square forms. Please follow Xiao Li's method to explore and solve the following problems:  
$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=(m+n\sqrt{3})^2$, express $a$ and $b$ in terms of $m$ and $n$: $a=$______, $b=$______;  
$(2)$ If $a+4\sqrt{3}=(m+n\sqrt{3})^2$, and $a$, $m$, $n$ are all positive integers, find the value of $a$;  
$(3)$ Simplify: $\sqrt{25+4\sqrt{6}}$.

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