
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 a regular triangular pyramid \(SABC\) with vertex \(S\), a height \(SD\) is dropped. Point \(K\) is taken on segment \(SD\) such that \(SK: KD = 1:2\). It is known that the dihedral angles between the base and the lateral faces are \(\frac{\pi}{6}\), and the distance from point \(K\) to the lateral edge is \(\frac{4}{\sqrt{13}}\). Find the volume of the pyramid.

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