$ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1, P_2, \dots, P_{10}$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \dots, Q_{10}$ for the side $CA$ and $R_1, R_2, \dots, R_{10}$ for the side $AB$. Find the number of triples $(i,j,k)$ with $i,j,k \in \{1,2,\dots,10\}$ such that the centroids of triangles $ABC$ and $P_iQ_jR_k$ coincide.

Let's think step by step and output the final answer within \boxed{}.
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