
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, Kang Kang 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, Kang Kang made the following exploration: Let $a+b\sqrt{2}={({m+n\sqrt{2}})^2}$ (where $a$, $b$, $m$, $n$ are all positive integers), then $a+b\sqrt{2}={m^2}+2{n^2}+2mn\sqrt{2}$ (rational numbers and irrational numbers correspondingly equal), therefore $a=m^{2}+2n^{2}$, $b=2mn$. In this way, Kang Kang found a method to transform the expression $a+b\sqrt{2}$ into a square form. Please follow Kang Kang's method to explore and solve the following problems:  
$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}={({c+d\sqrt{3}})^2}$, express $a$ and $b$ in terms of $c$ and $d$: $a=$______, $b=$______;  
$(2)$ If $7-4\sqrt{3}={({e-f\sqrt{3}})^2}$, and $e$, $f$ are both positive integers, simplify $7-4\sqrt{3}$;  
$(3)$ Simplify: $\sqrt{7+\sqrt{21-\sqrt{80}}}$.

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