
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 solution to question (1) before answering question (2):  
$(1)$ Given that $a$ and $b$ are rational numbers, and satisfy the equation $5-\sqrt{3}a=2b+\frac{2}{3}\sqrt{3}-a$, find the values of $a$ and $b$.  
Solution: Since $5-\sqrt{3}a=2b+\frac{2}{3}\sqrt{3}-a$,  
we have $5-\sqrt{3}a=(2b-a)+\frac{2}{3}\sqrt{3}$,  
thus $\left\{\begin{array}{l}2b-a=5\\-a=\frac{2}{3}\end{array}\right.$,  
which gives $\left\{\begin{array}{l}a=-\frac{2}{3}\\ b=\frac{13}{6}\end{array}\right.$.

$(2)$ Given that $x$ and $y$ are rational numbers, and satisfy the equation ${x}^{2}-2y-\sqrt{2}y=17-4\sqrt{2}$, find the value of $2x+y$.

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