
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
For a positive real number  $a$ , let  $C$  be the cube with vertices at  $(\pm a, \pm a, \pm a)$  and let  $T$  be the tetrahedron with vertices at  $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of  $T$  and  $C$  has volume  $ka^3$  for some  $k$ , find  $k$ .

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