Let \( A = \frac{2^2 + 3 \cdot 2 + 1}{3! \cdot 4!} + \frac{3^2 + 3 \cdot 3 + 1}{4! \cdot 5!} + \frac{4^2 + 3 \cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2 + 3 \cdot 10 + 1}{11! \cdot 12!} \). Find the remainder when \( 11! \cdot 12! \cdot A \) is divided by 11.

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