
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
Consider a rectangular parallelepiped \( A_{1} B_{1} C_{1} D_{1}-A B C D \), with edge lengths \( A A_{1}=a, B_{1} A_{1}=b, A_{1} D_{1}=c \). Points \( M \), \( N \), \( P \), and \( Q \) are the midpoints of \( A_{1} B_{1} \), \( A_{1} D_{1} \), \( B C \), and \( C D \), respectively. Determine the distance between the centroids of triangles \( A M N \) and \( C_{1} P Q \).

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