
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
On the line $y=-x+1$, for points $A(x_A,y_A)$ and $B(x_B,y_B)$, whose "inverse points" $A'\left( \frac{1}{x_A},\frac{1}{y_A} \right)$ and $B'\left( \frac{1}{x_B},\frac{1}{y_B} \right)$ are both on the graph of $y=\frac{k}{x}$, with $AB=2\sqrt{2}$, calculate the value of $k$.

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