
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

a) The vertices \(M\) and \(N\) of a cube with edge length \(a\) are symmetric with respect to the center of the cube. Find the length of the shortest path from \(M\) to \(N\) along the surface of the cube.

b) A box has the shape of a rectangular parallelepiped with dimensions \(30 \times 12 \times 12\). Point \(A\) is located on the face \(12 \times 12\), at a distance of 1 from one side of this face and equidistant from the two other parallel sides. What is the length of the shortest path on the surface of the box from point \(A\) to the point symmetric to it with respect to the center of the parallelepiped?

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