
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 $a+b=n$, then $a$ and $b$ are called "balanced numbers" about the integer $n$. For example, $3$ and $-4$ are "balanced numbers" about $-1$, and $2$ and $8$ are "balanced numbers" about $10$.  
$(1)$ Fill in the blank: $-6$ and $8$ are "balanced numbers" about ______;  
$(2)$ Given $a=6x^{2}-4kx+8$ and $b=-2(3x^{2}-2x+k)$ (where $k$ is a constant), and $a$ and $b$ are always "balanced numbers" about an integer $n$, regardless of the value of $x$, find the value of $n$.

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