
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 following text:  
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 part of $\sqrt{2}$, so Xiao Ming uses $\sqrt{2}-1$ to represent the decimal part of $\sqrt{2}$. Do you agree with Xiao Ming's representation? In fact, Xiao Ming's representation is reasonable. Since the integer part of $\sqrt{2}$ is $1$, subtracting the integer part from the number gives the decimal part. For example, since $\sqrt{4} \lt \sqrt{7} \lt \sqrt{9}$, that is, $2 \lt \sqrt{7} \lt 3$, the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$.  
Based on the above knowledge, answer the following questions:  
$(1)$ Given that the decimal part of $\sqrt{5}$ is $a$, and the integer part of $\sqrt{13}$ is $b$, find the value of $a+b+5$;  
$(2)$ Given $10+\sqrt{3}=x+y$, where $x$ is an integer and $0 \lt y \lt 1$, find the opposite of $x-y+\sqrt{3}$;  
$(3)$ Given that the decimal part of $5+\sqrt{11}$ is $a$, and the decimal part of $5-\sqrt{11}$ is $b$, find the value of $a+b$.

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