
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
Complete the following questions:  
$(1)$ Calculate: $(\sqrt{8}-\sqrt{\frac{1}{2}})\div \sqrt{2}$.  
$(2)$ Calculate: $2\sqrt{3}\times (\sqrt{12}-3\sqrt{75}+\frac{1}{3}\sqrt{108})$.  
$(3)$ Given $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, find the value of the algebraic expression $a^{2}-3ab+b^{2}$.  
$(4)$ Solve the equation: $\left(2x-1\right)^{2}=x\left(3x+2\right)-7$.  
$(5)$ Solve the equation: $2x^{2}-3x+\frac{1}{2}=0$.  
$(6)$ Given that real numbers $a$ and $b$ are the roots of the equation $x^{2}-x-1=0$, find the value of $\frac{b}{a}+\frac{a}{b}$.

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