
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
A student made a mistake when simplifying $\frac{1}{2}x-2(x-\frac{1}{3}y^{2})+(-\frac{3}{2}x+\frac{1}{3}y^{2})$. The solution process is as follows:  
Original expression $=\frac{1}{2}x-(2x-\frac{2}{3}y^{2})+(-\frac{3}{2}x+\frac{1}{3}y^{2})$ (Step 1)  
$=\frac{1}{2}x-2x-\frac{2}{3}y^{2}-\frac{3}{2}x+\frac{1}{3}y^{2}$ (Step 2)  
$=-3x-\frac{1}{3}y^{2}$ (Step 3)  
$(1)$ The student made a mistake starting from step ____, the reason for the error is ____.  
$(2)$ Write down the correct solution process.

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