
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
Points \( E \) and \( F \) are marked on the edges \( A A_1 \) and \( C C_1 \) of cube \( A B C D A_1 B_1 C_1 D_1 \), respectively, such that \( A E = 2 A_1 E \) and \( C F = 2 C_1 F \). A plane through points \( B, E, \) and \( F \) divides the cube into two parts. Find the ratio of the volume of the part containing point \( B_1 \) to the volume of the entire cube.

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