<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 The sequence of real numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{100}\) satisfies the inequalities \(a_{n} - 2022 a_{n+1} + 2021 a_{n+2} \geq 0\) for \(n = 1, 2, 3, \ldots, 98\), and \(a_{99} - 2022 a_{100} + 2021 a_{1} \geq 0\), \(a_{100} - 2022 a_{1} + 2021 a_{2} \geq 0\). Find \(a_{22}\), given that \(a_{10} = 10\). Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
10
The sequence of real numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{100}\) satisfies the inequalities \(a_{n} - 2022 a_{n+1} + 2021 a_{n+2} \geq 0\) for \(n = 1, 2, 3, \ldots, 98\), and \(a_{99} - 2022 a_{100} + 2021 a_{1} \geq 0\), \(a_{100} - 2022 a_{1} + 2021 a_{2} \geq 0\). Find \(a_{22}\), given that \(a_{10} = 10\).
test