A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January $1$. On what day does his unit of blood expire? Provide the answer by converting the month to its corresponding number (e.g., January is 1, February is 2, etc.) and the day to a number, then give the sum of these two numbers.

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