
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
【Given】:$\because \sqrt{4}＜\sqrt{6}＜\sqrt{9}$,<br/>$\therefore 2＜\sqrt{6}＜3$,<br/>$\therefore$ the integer part of $\sqrt{6}$ is $2$, and the decimal part is $\sqrt{6}-2$.<br/>【Application】:<br/>$(1)$ The integer part of $\sqrt{30}$ is ______, and the decimal part is ______;<br/>$(2)$ Given that the integer part of $6+\sqrt{15}$ is $x$, and the decimal part of $6-\sqrt{15}$ is $y$, find the value of $x+y$;<br/>【Extension】: Given that $a$ and $b$ are rational numbers, and $(a+\sqrt{3})^2=b-8\sqrt{3}$, find the value of $a-b$.

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