
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 that the domain of the function $f(x)$ is $R$, $f(2x-2)$ is an even function, $f(x-3)+f(-x+1)=0$, when $x\in [-2,-1]$, $f(x)=\frac{1}{a^x}-ax-4$ ($a>0$ and $a\neq 1$), and $f(-2)=4$, find $\sum_{k=1}^{19}{|{f(k)}|}$.

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