
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 \(ABCD\) is the base of a pyramid with apex \(S\). All its lateral faces are inclined to the base plane at an angle of \(60^{\circ}\). The 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 upper face of the parallelepiped (opposite to 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 points \(M, N, K, L, F, P, R, Q\) 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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