
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
$(1)$ The following is the process of Lele simplifying the polynomial. Please read carefully and complete the corresponding tasks.<br/>Solution: $6mn-2m-3\left(m+2mn\right)$<br/>$=6mn-2m-\left(3m+6mn\right)\ldots $First step,<br/>$=6mn-2m-3m+6mn\ldots $Second step,<br/>$=12mn-5m\ldots $Third step.<br/>Task one:<br/>① The operation law used in the first step of the simplification above is ______;<br/>② The error occurred in the ______ step of the above simplification, and the reason for the error is ______;<br/>Task two: Write down the correct simplified result ______;<br/>$(2)$ Simplify first, then evaluate: $a^2b^3-\frac{1}{2}(4ab+6a^2b^3-1)+2(ab-a^2b^3)$, where $a=\frac{1}{2},b=3$.

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