
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
Some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}={(1+\sqrt{2})}^{2}$. Let $a+b\sqrt{2}=(m+n\sqrt{2})^{2}$ (where $a$, $b$, $m$, $n$ are all positive integers), then we have $a+b\sqrt{2}=m^{2}+2n^{2}+2mn\sqrt{2}$, so $a=m^{2}+2m^{2}$, $b=2mn$. This method can be used to convert some expressions of the form $a+b\sqrt{2}$ into square form. Please explore and solve the following problems using the method described above:

$(1)$ 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 $n$: $a=$______, $b=$______;

$(2)$ Find a set of positive integers $a$, $b$, $m$, $n$ to fill in the blanks: ______$+$______$\sqrt{5}=( \_\_\_\_\_\_+\_\_\_\_\_\_\sqrt{5})^{2}$;

$(3)$ Simplify $\frac{1}{\sqrt{16-6\sqrt{7}}}-\frac{1}{\sqrt{11+4\sqrt{7}}}$.

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