
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 prism $A B C D A_1 B_1 C_1 D_1$ is the rectangle $A B C D$. The acute angles $D_1 D A$ and $D_1 D C$ are equal to each other. The angle between the edge $D D_1$ and the plane of the base of the prism is $\arccos{1}$. All faces of the prism touch a certain sphere.

Given $CD = 5$, find $BC$ and the angle between the planes $D_1 D C$ and $A B C$, as well as the distance from point $D$ to the center of the sphere.

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