
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
Find the area of the cross-section of a regular hexagonal pyramid \( SABCDEF \) cut by a plane passing through vertex \( F \) of the base \( ABCDEF \) and parallel to the median \( CM \) of the lateral face \( SCD \) and the apothem \( SN \) of the lateral face \( SAF \), given that the side of the base of the pyramid is \( 4\sqrt{7} \), and the distance from the vertex \( S \) to the cutting plane is \( \sqrt{7} \). (20 points)

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