
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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

Present the answer in LaTex format: \boxed{Your answer}
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