
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 \(ABCD A_1B_1C_1D_1\) with side length \(a\), point \(K\) is the midpoint of edge \(B_1C_1\), point \(L\) lies on edge \(C_1D_1\) such that \(D_1L = 3C_1L\), and point \(N\) is the midpoint of edge \(AA_1\). Construct the intersection of the cube with the plane passing through points \(K\), \(L\), and \(N\). Describe the algorithm for constructing the section. Find the area of the resulting section.

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