
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
The telephone company "Prosto-Telecom" in Prostokvashino uses three-digit phone numbers. The equipment is old, so there may be errors in individual digits of the transmitted subscriber number — each digit can be independently replaced by another random digit with a probability of $p=0.02$. To reduce the likelihood of incorrect connections, the company uses the following rule: the first digit of the number must always be the remainder when the sum of the other two digits is divided by 10. For example, the numbers 000 and 156 are possible, but the number 234 is not. If the check digit is incorrect during the connection, an error message is issued. Find the probability that, despite the measures taken, an incorrect connection will occur.

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