
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
In the seventh-grade activity class, three students each took a card with a polynomial written on it, denoted as $A$, $B$, and $C$ respectively (as shown in the table below), where $C$ is unknown.

| $A=-2x^{2}-(k-1)x+1$ | $B=-2(x^{2}-x+2)$ | $C$ |

$(1)$ If $A$ is a quadratic binomial, then the value of $k$ is ______;  
$(2)$ When $k=-1$, $C+2A=B$, then $C=\_\_\_\_\_\_$.

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