
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
Definition: If $a+b=3$, then $a$ and $b$ are called peace numbers about $3$.  
$(1)$ $4$ and ______ are peace numbers about $3$, $8-x$ and ______ are peace numbers about $3$. (Fill in an algebraic expression containing $x$)  
$(2)$ If $a=x^{2}-4x-1$, $b=x^{2}-2(x^{2}-2x-2)$, determine whether $a$ and $b$ are peace numbers about $3$, and explain the reason.  
$(3)$ If $c=kx+1$, $d=x-2$, and $c$ and $d$ are peace numbers about $3$, if $x$ is a positive integer, find the value of the non-negative integer $k$.

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