
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
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  $a + b$ .

*Proposed by Evan Chang*

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