
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
In the vertices \( A \), \( B \), and \( C \) of an equilateral triangle \( ABC \) with side length 1, perpendiculars are drawn to its plane, and points \( A_1 \), \( B_1 \), and \( C_1 \) are taken on these perpendiculars, all on the same side of the plane \( ABC \). Given that \( A A_1 = 4 \), \( B B_1 = 5 \), and \( C C_1 = 6 \), find the volume of the polyhedron \( ABC A_1 B_1 C_1 \).

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