
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 the parabola $y^2=2px$ has its focus $F$ coinciding with the right focus of the hyperbola $\frac{x^2}{7} - \frac{y^2}{9} = 1$, and the directrix of the parabola intersects the x-axis at point $K$. If point $A$ is on the parabola and $|AK| = \sqrt{2}|AF|$, then the area of $\triangle AFK$ is \_\_\_\_\_\_.

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