
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
Let \( A B C D A_{1} B_{1} C_{1} D_{1} \) be a given cube with side length 1. Connect point \( K \) with the midpoint of edge \( A_{1} A \) at point \( M \). Then \( D F = F A = \frac{1}{2} \). Let \( N \) be the midpoint of \( C C_{1} \). Extend \( F M \) until it intersects \( D_{1} A_{1} \) at point \( O \). Since the cutting plane passes through \( N M \parallel C A \) and \( N M \parallel A_{1} C_{1} \), it intersects the base \( D C B A \) along a line parallel to \( C A \). Therefore, \( F L \), where \( L \) is the midpoint of \( D C \), lies in the cutting plane. From point \( N \), draw a line parallel to \( F M \) which intersects \( C_{1} B_{1} \) at some point \( M_{1} \). The segment \( M_{1} O \) lies in the plane of the top base. Thus, \( M_{1} O \) intersects \( A_{1} B_{1} \) at the midpoint of \( A_{1} B_{1} \) at some point \( P \), which divides \( A_{1} B_{1} \) into halves. Connecting point \( P \) with point \( M \), we obtain the desired section of the cube. It is clear that this section is a regular hexagon with side length \(\frac{1}{\sqrt{2}}\). Therefore, the area of the section is \(6 \left(\frac{1}{\sqrt{2}}\right)^{2} \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{4}\).

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