
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
The rhombus \(ABCD\) forms the base of a pyramid with the apex \(S\). All lateral faces of the pyramid are inclined to the base plane at an angle of \(60^\circ\). Points \(M, N, K,\) and \(L\) are the midpoints of the sides of the rhombus \(ABCD\). A rectangular parallelepiped is constructed on the rectangle \(MNKL\) as its base. The edges of the top face of the parallelepiped (opposite the face \(MNKL\)) intersect the lateral edges of the pyramid \(SABCD\) at points \(F, P, R,\) and \(Q\), respectively. The volume of the polyhedron with vertices at \(M, N, K, L, F, P, R, Q\) is \(12\sqrt{3}\), and the radius of the circle inscribed in the rhombus \(ABCD\) is \(2.4\). Find the side length of the rhombus \(ABCD\).

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