
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 diagonals of a rectangular parallelepiped \( ABCDA_1B_1C_1D_1 \), inscribed in a sphere of radius \( R \), are inclined at an angle of \( 45^\circ \) to the plane of the base. Find the area of the cross-section of this parallelepiped by a plane that passes through the diagonal \( AC_1 \), is parallel to the diagonal \( BD \) of the base, and forms an angle of \( \arcsin \frac{\sqrt{2}}{4} \) with the diagonal \( BD_1 \).

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