
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
Given a right prism \( ABCD A_1 B_1 C_1 D_1 \) with base \( ABCD \). Planes \(\alpha\) and \(\beta\) are perpendicular to \( B_1D \) and pass through vertices \( A \) and \( D_1 \) respectively. Let \( F \) and \( H \) be the points of intersection of planes \(\alpha\) and \(\beta\) with the diagonal \( B_1D \), such that \( DF < DH \).

a) Find the ratio \( B_1H : DF \).

b) Additionally, it is known that a sphere with radius 3 touches all the lateral faces of the prism, as well as planes \(\alpha\) and \(\beta\). Find the segment \( B_1D \) and the volume of the prism \( ABCD A_1 B_1 C_1 D_1 \).

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