
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】When studying quadratic radicals, Xiaoming found that some expressions containing radicals can be simplified into the square of another expression, such as:  
$5+2\sqrt{6}=(2+3)+2\sqrt{2×3}=(\sqrt{2})^2+(\sqrt{3})^2+2\sqrt{2}×\sqrt{3}=(\sqrt{2}+\sqrt{3})^2$;  
$8+2\sqrt{7}=(1+7)+2\sqrt{1×7}=1^{2}+(\sqrt{7}\;)^{2}+2\times 1\times \sqrt{7}=(1+\sqrt{7})^{2}$.  
【Analogical Induction】  
$(1)$ Please imitate Xiaoming's method to simplify $7+2\sqrt{10}$ into the square of another expression;  
$(2)$ Please simplify $\sqrt{11-6\sqrt{2}}$ using Xiaoming's method.  
【Variation Exploration】  
$(3)$ If $a+2\sqrt{21}=(\sqrt{m}+\sqrt{n})^{2}$, and $a$, $m$, $n$ are all positive integers, find the value of $a$.

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