
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
Which quadrilateral pyramids have faces consisting only of equilateral triangles and isosceles right triangles? If there is more than one of each type of triangle, they must be congruent.

Calculate the volume of the pyramid with the maximum number of isosceles right triangles. Use the shortest side as the unit of measurement.

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