
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
$(1)$ Calculate: $(2\sqrt{3}+\sqrt{6})(2\sqrt{3}-\sqrt{6})-(\sqrt{2}-1)^{2}$;  
$(2)$ For a quadratic equation $ax^{2}+bx+c=0$ ($a$, $b$, $c$ are constants, $a\neq 0$), when $b^{2}-4ac\geq 0$, please derive the formula for finding the roots of the equation using the completing the square method.

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