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Problem 867: Tiling Dodecagon

There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.

Let $T(n)$ be the number of ways to tile a regular dodecagon of side $n$ with regular polygons of side 1. Then $T(1) = 5$. You are also given $T(2) = 48$.

Find $T(10)$. Give your answer modulo $10^9 + 7$.