
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
Read the text below and answer the questions. Everyone knows that $\sqrt{2}$ is an irrational number, and irrational numbers are infinite non-repeating decimals. Therefore, we cannot write out all the decimal parts of $\sqrt{2}$, but since $1 \lt \sqrt{2} \lt 2$, the integer part of $\sqrt{2}$ is $1$. Subtracting the integer part $1$ from $\sqrt{2}$ gives the decimal part as $(\sqrt{2}-1)$. Answer the following questions:  
$(1)$ The integer part of $\sqrt{10}$ is ______, and the decimal part is ______;  
$(2)$ If the decimal part of $\sqrt{6}$ is $a$, and the integer part of $\sqrt{13}$ is $b$, find the value of $a+b-\sqrt{6}$;  
$(3)$ Given $12+\sqrt{3}=x+y$, where $x$ is an integer and $0 \lt y \lt 1$, find the opposite of $x-y$.

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