<|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
 Katelyn is building an integer (in base  $10$ ). She begins with  $9$ . Each step, she appends a randomly chosen digit from  $0$  to  $9$  inclusive to the right end of her current integer. She stops immediately when the current integer is  $0$  or  $1$  (mod  $11$ ). The probability that the final integer ends up being  $0$  (mod  $11$ ) is  $\tfrac ab$  for coprime positive integers  $a$ ,  $b$ . Find  $a + b$ . 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
31
Katelyn is building an integer (in base  $10$ ). She begins with  $9$ . Each step, she appends a randomly chosen digit from  $0$  to  $9$  inclusive to the right end of her current integer. She stops immediately when the current integer is  $0$  or  $1$  (mod  $11$ ). The probability that the final integer ends up being  $0$  (mod  $11$ ) is  $\tfrac ab$  for coprime positive integers  $a$ ,  $b$ . Find  $a + b$ .
test