
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
In Vovochka's method of adding numbers, he does not carry over digits. Instead, he writes down the sum of each pair of digits in their respective place values, even if the result is a two-digit number. For example, for the sum $248+208$, Vovochka would get 4416.

a) In how many cases will Vovochka get the correct answer when adding all possible pairs of three-digit numbers? (If Vovochka has already added two different numbers in a different order previously, he does not notice it.)

b) Find the smallest possible difference between the correct answer and Vovochka's answer for all other pairs of three-digit numbers.

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