
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
Simplify: $\sqrt{7+4\sqrt{3}}$, using the following method:  
First, rewrite $\sqrt{7+4\sqrt{3}}$ as $\sqrt{7+2\sqrt{12}}$. Since $4+3=7$ and $4\times3=12$,  
we have $(\sqrt{4})^2+(\sqrt{3})^2=7$ and $\sqrt{4}\times\sqrt{3}=\sqrt{12}$,  
thus $\sqrt{7+4\sqrt{3}}=\sqrt{7+2\sqrt{12}}=\sqrt{(\sqrt{4})^2+(\sqrt{3})^2+2\sqrt{4}\times\sqrt{3}}=\sqrt{(\sqrt{4}+\sqrt{3})^2}$  
$=\sqrt{4}+\sqrt{3}=2+\sqrt{3}$.  
Using the method above, simplify: $\sqrt{12-4\sqrt{5}}=$ \_\_\_\_\_\_.

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