
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
Given a positive integer  $n$ , we define  $\lambda (n)$  as the number of positive integer solutions of  $x^2-y^2=n$ . We say that  $n$  is *olympic* if  $\lambda (n) = 2021$ . Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?

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