
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 \\(Γ\\): \\(y^{2} = 8x\\) with focus \\(F\\), and the directrix intersects the \\(x\\)-axis at point \\(K\\). If point \\(P\\) is on \\(Γ\\) and \\(\left| PK \right| = \sqrt{2}\left| PF \right|\\), then the area of \\(\Delta PKF\\) is \_\_\_\_\_\_\_\_\_\_\_\_\_\_.

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