
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
Inside a rectangular parallelepiped \( A B C D A_1 B_1 C_1 D_1 \), there are two spheres, \( \sigma_1 \) and \( \sigma_2 \), that touch each other externally. Additionally, sphere \( \sigma_1 \) touches the faces \( A B C D \), \( A B B_1 A_1 \), and \( A D D_1 A_1 \), while sphere \( \sigma_2 \) touches the faces \( A_1 B_1 C_1 D_1 \), \( B C C_1 B_1 \), and \( C D D_1 C_1 \). It is known that \( A B = 6 - \sqrt{2} \), \( A_1 D_1 = 6 + \sqrt{2} \), and \( C C_1 = 6 \). Find the distance between the centers of the spheres \( \sigma_1 \) and \( \sigma_2 \). Find the minimum and maximum total volume of the spheres.

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