
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
Given: $A=ax^{2}-x-1$, $B=3x^{2}-2x+2$ (where $a$ is a constant).  
$(1)$ When $a=\frac{1}{2}$, simplify $B-2A$;  
$(2)$ Under the condition in $(1)$, if $B-2A-2C=0$, find $C$;  
$(3)$ If the sum of $A$ and $B$ does not contain the term $x^{2}$, find the value of $a$.

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