
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 a rectangular parallelepiped \(A B C D A_1 B_1 C_1 D_1\) (where \(A B C D\) and \(A_1 B_1 C_1 D_1\) are the bases, and \(A A_1 \parallel B B_1 \parallel C C_1 \parallel D D_1\)), the segments \(M_1 N_1, M_2 N_2, M_3 N_3\) are common perpendiculars to the pairs of segments \(A_1 C_1\) and \(A B_1\), \(B C_1\) and \(A C\), \(D C_1\) and \(A D_1\) respectively. The volume of the parallelepiped is \(V\), the radius of the circumscribed sphere is \(R\), and the sum of the lengths of the edges \(A A_1\), \(A B\), and \(A D\) is \(m\). Find the sum of the volumes of the pyramids \(A A_1 M_1 N_1\), \(A B M_2 N_2\), and \(A D M_3 N_3\).

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