
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]


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Answer the following questions : $\textbf{(a)}$   Find all real solutions of the equation   $$ \Big(x^2-2x\Big)^{x^2+x-6}=1 $$   Explain why your solutions are the only solutions. $\textbf{(b)}$   The following expression is a rational number. Find its value.   $$ \sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10} $$  

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