Let $p = 9001$ be a prime number and let $\mathbb{Z}/p\mathbb{Z}$ denote the additive group of integers modulo $p$.  Furthermore, if $A, B \subset \mathbb{Z}/p\mathbb{Z}$, then denote $A+B = \{a+b \pmod{p} | a \in A, b \in B \}.$ Let $s_1, s_2, \dots, s_8$ are positive integers that are at least $2$. Yang the Sheep notices that no matter how he chooses sets $T_1, T_2, \dots, T_8\subset \mathbb{Z}/p\mathbb{Z}$ such that $|T_i| = s_i$ for $1 \le i \le 8,$ $T_1+T_2+\dots + T_7$ is never equal to $\mathbb{Z}/p\mathbb{Z}$, but $T_1+T_2+\dots+T_8$ must always be exactly $\mathbb{Z}/p\mathbb{Z}$. What is the minimum possible value of $s_8$?

[i]Proposed by Yang Liu
1. We are given that \( p = 9001 \) is a prime number and we are working in the additive group \(\mathbb{Z}/p\mathbb{Z}\).
2. We need to find the minimum possible value of \( s_8 \) such that for any sets \( T_1, T_2, \dots, T_8 \subset \mathbb{Z}/p\mathbb{Z} \) with \( |T_i| = s_i \) for \( 1 \le i \le 8 \), the sum \( T_1 + T_2 + \dots + T_8 \) covers all elements of \(\mathbb{Z}/p\mathbb{Z}\), i.e., \( T_1 + T_2 + \dots + T_8 = \mathbb{Z}/p\mathbb{Z} \).

3. We are also given that \( T_1 + T_2 + \dots + T_7 \) is never equal to \(\mathbb{Z}/p\mathbb{Z}\). This implies that the product of the sizes of the sets \( T_1, T_2, \dots, T_7 \) is less than \( p \):
   \[
   \prod_{i=1}^7 s_i < p
   \]

4. To find the minimum \( s_8 \), we need to ensure that the product of the sizes of the sets \( T_1, T_2, \dots, T_8 \) is at least \( p \):
   \[
   \prod_{i=1}^8 s_i \geq p
   \]

5. Let's consider the smallest possible values for \( s_1, s_2, \dots, s_7 \). Since each \( s_i \) is at least 2, we can set \( s_1 = s_2 = \dots = s_6 = 2 \) and \( s_7 = 140 \) (since \( 2^6 \times 140 = 128 \times 140 = 17920 \), which is greater than \( 9001 \)).

6. Now, we need to find \( s_8 \) such that:
   \[
   2^6 \times 140 \times s_8 \geq 9001
   \]
   Simplifying, we get:
   \[
   128 \times 140 \times s_8 \geq 9001
   \]
   \[
   17920 \times s_8 \geq 9001
   \]
   \[
   s_8 \geq \frac{9001}{17920}
   \]
   \[
   s_8 \geq 0.5028
   \]

7. Since \( s_8 \) must be an integer and at least 2, the minimum possible value for \( s_8 \) is 2.

The final answer is \( \boxed{2} \).