
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. 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 $\sqrt{2}$. 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 method is reasonable because the integer part of $\sqrt{2}$ is $1$, subtracting this number from its integer part gives the decimal part. Please answer the following:<br/>$(1)$ If the integer part of $\sqrt{13}$ is denoted as $a$ and the decimal part as $b$, find the value of $a^{2}+b-\sqrt{13}$.<br/>$(2)$ Given: $10+\sqrt{3}=x+y$, where $x$ is an integer and $0 \lt y \lt 1$, find the value of $x-y$.

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