
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
Consider the equation $\frac{3x^2}{x-2} - \frac{3x+8}{4} + \frac{5-9x}{x-2} + 2 = 0$. Simplify and solve for $x$, considering the restrictions imposed by the domain.
**A)** $\frac{5+\sqrt{5}}{3}$ and $\frac{5-\sqrt{5}}{3}$  
**B)** Only $\frac{5+\sqrt{5}}{3}$  
**C)** Only $\frac{5-\sqrt{5}}{3}$  
**D)** Neither $\frac{5+\sqrt{5}}{3}$ nor $\frac{5-\sqrt{5}}{3}$  
**E)** $2$ and some other root

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