
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
Two opposite lateral faces of a quadrilateral pyramid $S A B C D$ are perpendicular to the base, and the height of the pyramid is $\sqrt{5}$. The base of the pyramid is an isosceles trapezoid $A B C D$ ($A D = B C$) inscribed in a circle such that $A B = 6$ and $\angle B A D = \frac{\pi}{3}$. Find the distance from point $D$ to the plane $S A B$. Inside the pyramid, there is a cone such that the circle of its base is inscribed in the triangle $S C D$, and its apex lies on the face $S A B$. Find the volume of the cone.

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