
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 base of the triangular pyramid \(ABCD\) is an equilateral triangle \(ABC\). The face \(BCD\) forms a \(60^\circ\) angle with the base plane. On the line passing through point \(D\) perpendicular to the base, lies the center of a sphere with unit radius, which touches the edges \(AB\), \(AC\), and the face \(BCD\). The height of the pyramid \(DH\) is half the length of a side of the base. Find the volume of the pyramid.

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