
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 (1) first, then answer question (2):  
$(1)$ Given that $a$ and $b$ are rational numbers, and satisfy the equation $2b+\sqrt{3}a=a+5-2\sqrt{3}$, find the values of $a$ and $b$;  
Solution:  
Since $2b+\sqrt{3}a=a+5-2\sqrt{3}$, then $2b-a+\sqrt{3}a=5-2\sqrt{3}$,  
which means $\left(2b-a\right)+\sqrt{3}a=5-2\sqrt{3}$.  
Also, since $a$ and $b$ are rational numbers, $2b-a$ is also a rational number,  
thus $\left\{{\begin{array}{l}{2b-a=5}\\{a=-2}\end{array}}\right.$ gives $\left\{{\begin{array}{l}{a=-2}\\{b=\frac{3}{2}}\end{array}}\right.$.

$(2)$ Given that $m$ and $n$ are rational numbers, and $m$ and $n$ satisfy the equation $m+2n+\sqrt{2}\left(2-n\right)=\sqrt{2}(\sqrt{2}+6)+15$, find the cube root of ${(\sqrt{m}+n})^{100}$.

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