
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
We say that a positive integer is "cool" if, by adding the squares of its digits and repeating this operation successively, we obtain the number 1. For example, 1900 is cool because

$$
1900 \rightarrow 82 \rightarrow 68 \rightarrow 100 \rightarrow 1
$$

a) Find two consecutive two-digit cool numbers.

b) Verify that for every \( n > 2020 \), there exist two consecutive \( n \)-digit numbers that are cool.

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