
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 and answer the questions:<br/>We all know that $\sqrt{2}$ is an irrational number, and irrational numbers are infinite non-repeating decimals, so we cannot write out all the decimal parts of $\sqrt{2}$. Therefore, Xiao Ming uses $\sqrt{2}-1$ to represent the decimal part of $\sqrt{2}$. Do you agree with Xiao Ming's representation method?<br/>In fact, Xiao Ming's representation method is reasonable because the integer part of $\sqrt{2}$ is $1$, subtracting this number from $\sqrt{2}$ gives the decimal part.<br/>For example: Since $\sqrt{4}＜\sqrt{7}＜\sqrt{9}$, that is, $2＜\sqrt{7}＜3$,<br/>Therefore, the integer part of $\sqrt{7}$ is $2$, and the decimal part is $(\sqrt{7}-2)$.<br/>Please answer:<br/>$(1)$ If the decimal part of $\sqrt{13}$ is $a$, and the integer part of $\sqrt{29}$ is $b$, find the value of $a+b-\sqrt{13}$;<br/>$(2)$ 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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