
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
A regular triangular pyramid is dissected by a plane perpendicular to the base, which divides two sides of the base in half. Determine the volume of the truncated pyramid if the side of the base of the original pyramid is $a$, and the dihedral angle at the base measures $45^{\circ}$.

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