
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

In cube \( ABCD A_1 B_1 C_1 D_1 \) with side length \( a \), point \( K \) is the midpoint of edge \( B_1 C_1 \), point \( L \) lies on edge \( C_1 D_1 \) such that \( D_1 L = 2 C_1 L \), and point \( N \) is the midpoint of edge \( A A_1 \). Construct the cross section of the cube with a plane passing through points \( K, L, N \). Describe the algorithm for constructing the cross section. Find the area of the resulting section.

(20 points)

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