
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, so we cannot write out all the decimal part of $\sqrt{2}$. Therefore, Xiaoming uses $\sqrt{2}-1$ to represent the decimal part of $\sqrt{2}$. Do you agree with Xiaoming's representation? In fact, Xiaoming's representation method is reasonable because the integer part of $\sqrt{2}$ is $1$, subtracting this number from $\sqrt{2}$ gives the decimal part. For example, $\sqrt{4}＜\sqrt{5}＜\sqrt{9}$, that is, $2＜\sqrt{5}＜3$, so the integer part of $\sqrt{5}$ is $2$, and the decimal part is $\sqrt{5}-2$. Please answer the following questions based on the above information:<br/>$(1)$ The integer part of $\sqrt{34}$ is ______, and the decimal part is ______;<br/>$(2)$ If the integer part of $\sqrt{11}$ is $a$, and the integer part of $\sqrt{17}$ is $b$, find the cube root of $12a+7b$;<br/>$(3)$ Given $9-\sqrt{7}=x-y$, 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}
user