
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 a right parallelepiped \( ABCD A_1B_1C_1D_1 \) is a square \( ABCD \) with a side length of 1. The length of each of the lateral edges \( AA_1, BB_1, CC_1, DD_1 \) is \(\sqrt{3}\). A right circular cylinder is positioned such that the points \( A, A_1, \) and \( D \) lie on its lateral surface, and the cylinder's axis is parallel to the diagonal \( BD_1 \) of the parallelepiped. Find the radius of the cylinder.

An answer to this problem is given as \(\frac{4 \sqrt{2}}{5}\).

Two equal spheres \( S_1 \) and \( S_2 \) touch each other, and additionally, each sphere touches both faces \( P \) and \( Q \) of a right dihedral angle. Sphere \( S_1 \) touches face \( P \) at point \( A \). Through this point, a line is drawn which intersects sphere \( S_1 \) at point \( B \), touches sphere \( S_2 \) at point \( C \), and intersects face \( Q \) at point \( D \). The line \( AD \) forms an angle of \( 30^\circ \) with face \( P \). Find the ratio \( AB : BC : CD \).

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