
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
The apothem of the regular pyramid \( S A B C D \) is 2, and the lateral edge forms an angle with the base \( A B C D \) equal to \( \arctan \sqrt{} \). Points \( E, F, K \) are chosen on the edges \( A B, A D, S C \) respectively, such that \( \frac{A E}{E B} = \frac{A F}{F D} = \frac{S K}{K C} = \frac{1}{2} \). Find: 
1) the area of the section of the pyramid cut by the plane \( E F K \);
2) the distance from point \( D \) to the plane \( E F K \);
3) the angle between the line \( S D \) and the plane \( E F K \).

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