
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
We know that $\sqrt{2}$ is an irrational number, with an integer part of $1$. Therefore, Xiaoming represents the decimal part of $\sqrt{2}$ as $\sqrt{2}-1$. Please answer the following questions:<br/>$(1)$ If the decimal part of $\sqrt{5}$ is $a$, and the integer part of $\sqrt{13}$ is $b$, find the value of $a+b-\sqrt{5}$;<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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